PHY4604
R. D. Field
Department of Physics
Chapter4_17.doc
University of Florida
Special Unitary 2×2 Matrices
Consider the unitary matrix
−
=
*
*
)
,
(
a
b
b
a
b
a
U
with
a
2
+ b
2
= 1
.
Such a matrix is said to be “unimodular” since
det(U) = a
2
+ b
2
= 1
.
Two Component Spinors:
This “special” (
i.e.
unimodular) unitary
2×2
matrix operates on two component “spinors”
=
2
1
~
v
v
v
as follows
−
=
2
1
*
*
2
1
'
'
v
v
a
b
b
a
v
v
.
The transformation leaves the
norm of states invariant
since
>
<
=
+
=
+
+
=
+
=
>=
<
v
v
v
v
v
v
v
v
v
v
b
a
v
v
v
v
v
v
v
v

)
(




)



)(



(

'


'

'
'
)
'
'
(
'

'
2
1
*
2
*
1
2
2
2
1
2
2
2
1
2
2
2
2
2
1
2
1
*
2
*
1
Definition of a Group:
A collection of objects is said to form a
“group”
if
(1)
There exists
“group multiplication” and the product of any two
group members produces another member of the group (
i.e.
G
3
=
G
1
G
2
“closed” under group multiplication).
(2)
There exists an “identity”,
I
, such that
GI = IG = G
.
(3)
There exists an “inverse”,
G
1
, such that
G
1
G = GG
1
= 1
.
(4)
The group multiplication is “associative”:
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 Spring '07
 FIELDS
 mechanics, Complex number, Euler angles, group multiplication

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