7.3 Interactions of a Two-Component Spinor
171
This result can also be recast in matrix form as
H =
|α| β
H
α|
β|
(7.24)
or, inversely, as
H =
α|
β|
H
|α| β
(7.25)
To establish the connection between the spinor and the vector, we now need to verify
how transformations in the spinor are manifested as transformations in the vector.
Consider a finite unitary transformation of the spinor. The transformation belongs
to the unitary group, U(2), and, as we have seen, the determinant of this matrix is
unimodular. We consider the special case, however, where the determinant is +1.
Such matrices form the special unitary group, SU(2). The most general form of an
SU(2) matrix involves two complex parameters, say a and b, subject to the condition
that their squared norm, |a| 2 +|b| 2 , equals unity. These parameters are also known
as the Cayley–Klein parameters. (Cf. Problem 2.1.) One has
U =
ab
− ¯
b ¯
a
(7.26)
The operation ˆ
R transforms the spinor as follows:
|α ′ | β ′
= ˆ
R
|α| β
=
|α| β
U(R)
(7.27)
In order to apply the transformation to the interaction operator, we must also consider the effect of ˆ
R on the column of bra-functions. This simply requires the inverse
of the matrix, which, for a unitary matrix, is nothing but its complex conjugate transposed:
α ′ |
β ′ |
= ˆ
R
α|
β|
= ¯
U
T (R)
α|
β|
(7.28)
The transformation of the spinor thus changes the interaction matrix as follows:
H
′ =
α ′ |
β ′ |
H
|α ′ | β ′
= ¯
U
T
α|
β|
H
|α| β
U
= ¯
U
T × H × U
(7.29)
The transformed Hamiltonian matrix is defined by a new set of parameters
(x ′ ,y ′ ,z ′ ):
H
′ =
z ′
x ′ − iy ′
x ′ + iy ′
−z ′
(7.30)
171
This result can also be recast in matrix form as
H =
|α| β
H
α|
β|
(7.24)
or, inversely, as
H =
α|
β|
H
|α| β
(7.25)
To establish the connection between the spinor and the vector, we now need to verify
how transformations in the spinor are manifested as transformations in the vector.
Consider a finite unitary transformation of the spinor. The transformation belongs
to the unitary group, U(2), and, as we have seen, the determinant of this matrix is
unimodular. We consider the special case, however, where the determinant is +1.
Such matrices form the special unitary group, SU(2). The most general form of an
SU(2) matrix involves two complex parameters, say a and b, subject to the condition
that their squared norm, |a| 2 +|b| 2 , equals unity. These parameters are also known
as the Cayley–Klein parameters. (Cf. Problem 2.1.) One has
U =
ab
− ¯
b ¯
a
(7.26)
The operation ˆ
R transforms the spinor as follows:
|α ′ | β ′
= ˆ
R
|α| β
=
|α| β
U(R)
(7.27)
In order to apply the transformation to the interaction operator, we must also consider the effect of ˆ
R on the column of bra-functions. This simply requires the inverse
of the matrix, which, for a unitary matrix, is nothing but its complex conjugate transposed:
α ′ |
β ′ |
= ˆ
R
α|
β|
= ¯
U
T (R)
α|
β|
(7.28)
The transformation of the spinor thus changes the interaction matrix as follows:
H
′ =
α ′ |
β ′ |
H
|α ′ | β ′
= ¯
U
T
α|
β|
H
|α| β
U
= ¯
U
T × H × U
(7.29)
The transformed Hamiltonian matrix is defined by a new set of parameters
(x ′ ,y ′ ,z ′ ):
H
′ =
z ′
x ′ − iy ′
x ′ + iy ′
−z ′
(7.30)