G()&,
)
=
«0'
exp
I
me+i
(4v)4
~
„&
w(1
—
v'))
1(1+v
1
—
v
)
dv
dw
xi
+
2(JI+vf
JI
—
v)
21
—
v'
w'
dv
f
~p4
I'
(4&r)4
"
g
1
—
v'
~
w'
2i
(
Xexp~
iw+i
(.
(2.
35)
w(1
—
v')
)
=
i(4&r)
'«04
(
«09,
w(1
—
v')
)
w)w(
(1
v2)2
(2.
36)
The
integral
representation
(
k„'
'
(dk)
exp(ik„(x„—
x„'))
exp~
i
w(1
—
v')
~
~
J
(
4~g'
J
UI.
I
AN
SCH
W
I
N
GER
then
transforms
(2.
35)
into
G(li)
=
—
(dk)
exp(ik„(x„—
x„'))
8
(42r)
'
dzo
X
l
(1
—
v')dv
l
0
~n
k„2
Xcosi
1+
(1")
iw
(2
37)
4~02
The
Fourier
integral
contained
in
the
second
term
of
(2.
39)
can
be
recognized
as
related
to
that
of
the
function
h(x)
(Eq.
(A.
10)):
f
exp(ik„x„)
~(x)
=
P
il
(dk).
(2.
41)
(22r)
'
~
k„'+
a22
Indeed,
1
f
exp
(ik»x»)
(dk)
(2
)'
"
1+(k.
'/4
o')(1
—
v')
which
can
be
followed
by
an
integration
by
parts
with
respect
to
v,
according
to
(
v
)
f
dw
(
k»
"
dj
v
—
—
)
„—
cosi
ly
(1
—
v')
)w
3)
&2
w
E
4&22
so
that
16~22
(2.
42)
(1
—
v')'
((1
—
v2)&
)
2
f
cosw
k»
f'
(
v
dw
—
I
1
—
iv
dv
3
"2
w
2222
"o
&
3)
G(li)
=
1
log
8('x
—
x')
48m'
ym
0
(
k„'
dw
sini
1+
(1
—
v')
iw.
(2.
38)
0
4a22
The
first
I
integral
is
logarithmically
divergent
at
the
origin.
On
introducing
a
lower
limit,
mo,
we
obtain
1
—
(v'/3)
2
1
—
log
—
—
—
P
~
n2dv
3
yw2
2~22
&
2
1+
(k„2/42
')
(1
—
v')
as
the
value
of
the
integral
(2.
38),
where
y
=1.
781.
The
insertion
of
this
result
into
(2.
37)
yields
G(li)
=
1
1
log
8(x
—
x')
48m"
ymo
4
1
l'
(
V')
(
1
—
—
iv'dvP
(42r)'
~22
&,
E
3
)
exp
(ik„(x„—
x„')
)
X
~'
(dk)
(2.
39)
1+
(k„2/4K22)
(1
—
v'}
in
which
it
has
been
noticed
that
the
operator
'
is
equivalent
to
multiplication
by
—
k„'
in
its
effect
on
exp(ik„x„},
and
that
exp(ik„x„)
(dk)
=
b(x)
(22r)'
~
=
b(x2)
8(xi)
ii(x2)
b(x2).
(2.
40)
1
f'
(
2
(x
—
x')
i
42r2
~2
E(1
—
v2)&
p
w
9
—
v'dv.
(2.
43)
(1
v2)
2
T&
expression
for the
induced
current
that
is
obtained
from
this
form
for
G(li)
is
n
1
4
()„(x))=
—
—
log

J„(x)
—
—
u
l
d~'
3%
+MD
f'(
X
t
Zi
—
(x
—
x')
(
(
(1
—
v')
&
1.
—
'v2
3
X
v'd
v
J„(x'),
(—
2.
44)
(1
—
v')'
where
u=e2/42rkc
is
the
fine
structure
constant.
The
current
induced
at
a
given
point
is
thus
exhibited
in
two
parts:
a
logarithmically
diver
gent
multiple
of
the
external
current
at
that
point,
and
a
finite
contribution
involving
the
external
current
in
the
vicinity
of
the
given
point.
The
first
part
reduces
the
strength
of
the
external
current
by
a
constant
factor
and
hence
produces
an
unobservable
charge
renormaliza
tion,
as
discussed
in
I.
The
second
part
of
(2.
44)
is
therefore
the
physically
significant
induced
current.
QUANTUM
ELECTRODYNAMICS
663
An
alternative
form
for
the
latter
is'
t'
(j„(x)}=
a—
—
d(a'
Z~
(x
—
x')
I
vr
~
&0
((I
—
v')&
1
—
'v'
v'dv
"J„(x').
(2.
45)
(1
v2)
2
If
the
external
current
varies
sufhciently
slowly,
the
relevant
unit
of
length
being
1/~0=k/mac,
one
can
obtain
a
series
in
ascending
powers
of
'
applied
to
J„(x).
This
may
be
done
by
con
tinued
application
of
the
relation:
2
)
(1
—
v')'
k(1
—
v')&
j
16~0'
where
d
v'
here
denotes
a
threedimensional
volume
element.
Now
G(r)
=
&(r,
xo)dxo
(2.
49)
obeys
the
differential
equation
(j„(x))
=

—
a
dr'
'
dxo'
J,
jt'
2
2
XZ(
(r
—
r),
—
(x,
—
x,
')
)
E
(1
—
v')
(1
—
v')
1
—
—
v
v'dv
V"
J„(r'),
(2.
48)
(1
v2)
2
(V
—
~02)G(r)
=
—
8(r)
(2.
You've reached the end of your free preview.
Want to read all 29 pages?
 Fall '05
 Energy, Mass, Photon, Quantum Field Theory, Polarization, Electromagnetic field, J ULIAN, A„