172
7 Spherical Symmetry and Spins
In this way, the transformation of the spinor (|α|β) → (|α ′ |β ′ ) induces a transformation of the vector (x y z) → (x ′ y ′ z ′ ). In the vector space this transformation
is described by a matrix O(R). This matrix may easily be constructed by combining
the previous two equations. One has
x
′ y
′ z
′
= ˆ
R ( xyz ) = ( xyz ) O(R)
(7.31)
where the transformation matrix is given by
O(R) =
⎛
⎜
⎝
1
2 (a 2 +¯ a 2 − b 2 − ¯
b 2 ) −
i
2 (a 2 −¯ a 2 + b 2 − ¯
b 2 )
−ab − ab
i
2 (a 2 −¯ a 2 − b 2 + ¯
b 2 )
1
2 (a 2 +¯ a 2 + b 2 + ¯
b 2 ) −i(ab − ab)
a ¯
b +¯ ab
−i(a ¯
b −¯ ab)
|a| 2 −|b| 2
⎞
⎟
⎠
(7.32)
It can easily be shown that this matrix is an orthogonal transformation with determinant equal to unity. Hence, it belongs to the SO(3) group. As a result, it will leave
the squared length of the vector invariant:
x
2 + y
2 + z
2 = x
′ 2 + y
′ 2 + z
′ 2
(7.33)
This conservation of length is the property that confirms the previous identification
of the interaction matrix elements with a 3-vector and relates it to ordinary space. In
fact, by identifying the rotation matrices in Eqs. (7.3) and (7.32), we may determine
the Cayley–Klein parameters. Two solutions with opposite signs are possible:
a =
cos
α
2
− in z sin
α
2
b =
−n y sin
α
2
− in x sin
α
2
(7.34)
or
a =−
cos
α
2
− in z sin
α
2
b =−
−n y sin
α
2
− in x sin
α
2
(7.35)
As this equation shows, the mapping O(R) is not an isomorphism but a homomorphism (see Fig. 7.3). Indeed, the elements of the matrix O are bilinear in the a and
b parameters; hence, an overall sign change of the two Cayley–Klein parameters
will give the same rotational matrix. The mapping between SU(2) and SO(3) is a
two-to-one mapping. Each element of the rotation group in 3D space is the image
of two elements in SU(2). For this reason, SU(2) is also called a covering group
of SO(3). The unit element in SO(3) is the image of the identity matrix in SU(2)
and minus the identity matrix. This homomorphism also appears when we check
the parameter list of Eq. (7.4), which leaves the rotation matrix of the vector unchanged. The overall sign change of the rotation angle and the directional cosines to
7 Spherical Symmetry and Spins
In this way, the transformation of the spinor (|α|β) → (|α ′ |β ′ ) induces a transformation of the vector (x y z) → (x ′ y ′ z ′ ). In the vector space this transformation
is described by a matrix O(R). This matrix may easily be constructed by combining
the previous two equations. One has
x
′ y
′ z
′
= ˆ
R ( xyz ) = ( xyz ) O(R)
(7.31)
where the transformation matrix is given by
O(R) =
⎛
⎜
⎝
1
2 (a 2 +¯ a 2 − b 2 − ¯
b 2 ) −
i
2 (a 2 −¯ a 2 + b 2 − ¯
b 2 )
−ab − ab
i
2 (a 2 −¯ a 2 − b 2 + ¯
b 2 )
1
2 (a 2 +¯ a 2 + b 2 + ¯
b 2 ) −i(ab − ab)
a ¯
b +¯ ab
−i(a ¯
b −¯ ab)
|a| 2 −|b| 2
⎞
⎟
⎠
(7.32)
It can easily be shown that this matrix is an orthogonal transformation with determinant equal to unity. Hence, it belongs to the SO(3) group. As a result, it will leave
the squared length of the vector invariant:
x
2 + y
2 + z
2 = x
′ 2 + y
′ 2 + z
′ 2
(7.33)
This conservation of length is the property that confirms the previous identification
of the interaction matrix elements with a 3-vector and relates it to ordinary space. In
fact, by identifying the rotation matrices in Eqs. (7.3) and (7.32), we may determine
the Cayley–Klein parameters. Two solutions with opposite signs are possible:
a =
cos
α
2
− in z sin
α
2
b =
−n y sin
α
2
− in x sin
α
2
(7.34)
or
a =−
cos
α
2
− in z sin
α
2
b =−
−n y sin
α
2
− in x sin
α
2
(7.35)
As this equation shows, the mapping O(R) is not an isomorphism but a homomorphism (see Fig. 7.3). Indeed, the elements of the matrix O are bilinear in the a and
b parameters; hence, an overall sign change of the two Cayley–Klein parameters
will give the same rotational matrix. The mapping between SU(2) and SO(3) is a
two-to-one mapping. Each element of the rotation group in 3D space is the image
of two elements in SU(2). For this reason, SU(2) is also called a covering group
of SO(3). The unit element in SO(3) is the image of the identity matrix in SU(2)
and minus the identity matrix. This homomorphism also appears when we check
the parameter list of Eq. (7.4), which leaves the rotation matrix of the vector unchanged. The overall sign change of the rotation angle and the directional cosines to