E2
ax„"
"
ax„
i
1
1
X
—
j„(x)
—
jg(x')do),
'
4'[0].
(3.
30)

ULIAN
SCHNI
NGE14
%'e
have
thereby
succeeded
in
constructing
an
equation
of
motion
for
+[o]
which
no
longer
contains
the
electromagnetic
field
variables
in-
volved
in
the
supplementary
condition.
The
additional
term
thus
introduced
is
evidently
the
covariant
generalization
of
the
Coulomb
inter-
action
between
charges.
To
exhibit
the
latter
property
somewhat
more
clearly,
we
must
restrict
the
arbitrary
space-like
surface
r
to
a
plane
surface
with
the
normal
e„.
The
advantage
thereby
acquired
is
the
possi-
bility
of
asserting
that
and
i
8
i
8
V~
—
$(r,
0)
=-
—
D(r,
0)
=
—
B(r),
(3.
35)
c
Bt
c
Bt
1
8
—
—
$(r,
0)
=
c
Bt
4+r
(3.
36)
in
which
the
latter
statement
involves
the
con-
tent
of
the
equations
of
definition
(2.
17)
as
adapted
to
the
special
coordinate
system.
It
follows
from
(3.
35)
that
of
which
the
covariant
expression
is
(a
a)
+n„n,
~n(x
—
x')
=0,
&ax„"
ax„)
8
1
1
n„X)(x
—
x')
=-
n.
(x.
—
x.
')
=0
(3.
31)
ax„4
[(x„—
x„')']&
which
enables
(3.
30)
to
be
simplified,
yielding:
n„(x„—
x„')
=
0.
(3.
37)
B%'[0]
zoic
Bo
(x)
1
1
(
BX)(x
—
x')
—
-j„(x)
e„(x)
—
—
~
n.
The
energy-momentum
four-vector
is
modified
by
the
unitary
transformation
(3.
19),
according
to
1
X
—
n~„(x)-j),
(x')da),
'
4'[0].
(3.
32)
P
[0]~cig'[aj~
C~]c
—
iG'
f']
(3.
38)
The
evaluation
of
the
new
operator
P„[s]
in-
volves
the
following
transformations,
which
we
note
without
proof':
To
prove
(3.
31),
it
is
sufficient
to
verify
it
in
a
particular
coordinate
system.
It
is
always
pos-
sible
to
construct
a
reference
system
for
which
the
normal
to
a
plane
space-like
surface
is
di-
rected
along
the
time
axis;
in
other
words,
in
this
reference
system,
n„=(0,
0,
0,
i).
Equation
(3.31)
then
states
that
the
spatial
derivatives
of
$(r
—
r',
t
—
t')
vanish
for
t=t'.
This
will
be
true
if
X)(r,
0)
=0
for
all
r,
that
is,
if
$(r,
t)
is
an
odd
function
of
t.
New,
in
this
special
coordinate
system,
(3.
8)
becomes
e'c''jA
(x)e-'s't'
=A
(x)
t'(
a
+n„n.
I
Z)(x
—
x')-j,
(x')dir,
',
&.
&ax„ax„j
c
ay
ay
d~
4V.
—
kV.
s
""
2J.
"
"a
„a.
ay
ap
~ir,
kV.
—
kV.
2
g
Bxy
[email protected]
t'1
a)'
~
—
—
~
~(r,
t)
=
PX)(r,
t)
=D(r,
t)
(3.
33)
Ec
at)
1
aX'
1
as'-
+
—
~
d&v
—
gx
—
d&x
—
J)
kc,
c
Bxg
c
Bx,
(3.
39)
and,
since
(2.
17)
assures
us
that
D(r,
t)
is
an
odd
function
of
the
time,
the
necessary
property
of
S(r,
t)
is
established.
As
a
final
step,
we
note
In
virtue
of
the
supplementary
condition
(3.
26),
that,
in
this
special
coordinate
system
we
may
write
l9
1
8
n,
$(x
—
x')
=
—
—
S(r
—
r',
0),
QXp
c
8$
BA.
'(x)
q
A~(x)e[&]
=
(
&~(x)
—
]+[&].
(3
4o)
ax„
)
n„(x„—
x,
')
=0
(3.
34)
It
follows,
as
a
result
of
straightforward
simpli-

QUANTUM
ELECTRODYNAMICS
fication,
that
~
88'
88'
P,
t
o]=
—
der„
2cs»
y
Bxp,
Bx)g
88'
8
'
$8
8&,
)
'
8x„8x„&
8x,
)
8$
8f
+
—
I
do
1
1
1»
8n(x
x')—
+-
«
-j.
(x)~»(x)+-
~
&r
g4~
f
2~g
Bxt
1
1.
X
—
ng„(x)
—
jz(x')do),
',
(3.
41)
C
C
which
has
been
stated
as
an
operator
equation,
rather
than
a
derived
supplementary
condition,
with
the
understanding
that
the
operator
A
—
A'
shall
no
longer
appear
in
the
theory.

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- Fall '05
- Mass, Quantum Field Theory, commutation relations, Julian Schwinger, space-like surface