ex4-3, Fizyka, Kwantowa Teoria Pola

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1
Exercise 4.3 (10 pnts)
Using the momentum expansions of (x)
X
†
⇤
(x)=
(a
~
p
e
~
p
(x)+a
~
p
e
~
p
(x))
the total momentum becomes (normal ordering is implicit)
Z
P =
1
2
$
@
0
(ir)
d
3
x i
Z
=
1
2
X
X
$
@
0
(ir)
†
⇤
†
⇤
d
3
x
(a
~
q
e
~
q
(x)+a
~
q
e
~
q
(x))i
(a
~
p
e
~
p
(x)+a
~
p
e
~
p
(x))
|
{z
}
X
†
⇤
~p (a
~
p
e
~
p
(x) a
~
p
e
~
p
(x))
Z
=
1
2
X
$
@
0
~p (a
~
p
e
~
p
(x) a
†
⇤
†
⇤
d
3
x
(a
~
q
e
~
q
(x)+a
~
q
e
~
q
(x))i
~
p
e
~
p
(x)) ,
~
q ,
~
and using the orthonormality of e
~
p
(x),
Z
⇣
=
1
2
X
$
@
0
e
~
p
(x
)
$
@
0
e
†
†
⇤
⇤
d
3
x
a
~
q
a
~
p
e
~
q
(x)i
a
~
q
a
~
p
e
~
q
(x)i
~
p
(x)
|
{z
}
|
{z
}
~
q ,
~
! 0
! 0
⌘
$
@
0
e
$
@
0
e
~
p
(x)
†
⇤
†
⇤
a
~
q
a
e
~
q
(x)i
~
p
(x)
+a
~
q
a
~
p
e
~
q
(x)i
|
{z
}
|
{z
}
!
~
q , v
!
~
q , v
=
1
2
X
X
†
†
†
~p (: a
~
p
a
~
p
+ a
~
p
a
~
:)
=
~
p
a
~
.
|
{z
}
†
2a
~
p
a
~
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ex4-3

ex4-3, Fizyka, Kwantowa Teoria Pola

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