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(1)
The righthanded spinors
u
~
k,s
and
v
~
k,s
(
s
= 1
,
2) are given as follows in a frame where
k
= (
E,
0
,
0
,p
):
u
~
k,
1
=
±
√
E
+
p
0
²
,
v
~
k,
1
=
±
0
√
E

p
²
,
u
~
k,
2
=
±
0
√
E

p
²
,
v
~
k,
2
=
±

√
E
+
p
0
²
.
Prove the identity
2
X
s
=1
u
~
k,s
u
†
~
k,s
=
2
X
s
=1
v
~
k,s
v
†
~
k,s
=
k
·
¯
σ
for
~
k
in a general direction in two steps:
(i) Show that the identity holds for
k
= (
E,
0
,
0
,p
).
(ii) Derive the identity for general
k
by performing an appropriate rotation that brings
k
= (
E,
0
,
0
,p
) to
k
= (
E,p
sin
θ
cos
φ,p
sin
θ
sin
φ,p
cos
θ
).
(2)
The lefthanded spinors
u
~
k,s
and
v
~
k,s
(
s
= 1
,
2) are given as follows in a frame where
k
= (
E,
0
,
0
,p
):
u
~
k,
1
=
±
√
E

p
0
²
,
v
~
k,
1
=
±
0
√
E
+
p
²
,
u
~
k,
2
=
±
0
√
E
+
p
²
,
v
~
k,
2
=
±

√
E

p
0
²
.
Prove the identities
(i)
2
X
s
=1
u
~
k,s
v
T
~
k,s
=
m
i
σ
2
(ii)
2
X
s
=1
u
~
k,s
u
†
~
k,s
=
2
X
s
=1
v
~
k,s
v
†
~
k,s
=
k
·
σ
for
~
k
in a general direction.
1
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This note was uploaded on 12/14/2011 for the course PHY 5667 taught by Professor Okui during the Fall '10 term at FSU.
 Fall '10
 Okui

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