6 p-Wave Superconductivity and d-Vector Representation
199
6.11 Proofs and Exercise Solutions
6.11.1 Proof of the Cayley–Klein Relation
Proof
R (a·σ )R −
=
cos /2 1 − i sin /2 ˆ
·σ
(a·σ )
cos /2 1 + i sin /2 ˆ
·σ
=
cos /2 1 − i sin /2 ˆ
·σ
cos /2 (a·σ )
+i sin /2
(a · ˆ
)1 + i(a ∧ ˆ
)·σ
= cos
2
/2 (a·σ ) + i cos /2 sin /2 (a · ˆ
)1
− cos /2 sin /2 (a ∧ ˆ
) · σ
− i cos /2 sin /2
a · ˆ
1 − i(a ∧ ˆ
)·σ
+ sin
2
/2
(a · ˆ
)( ˆ
·σ ) + i
2
ˆ
∧ (a ∧ ˆ
)
·σ
= (a · ˆ
)( ˆ
·σ ) + cos
2
/2
a − (a · ˆ
) ˆ
·σ + sin (a ∧ ˆ
)·σ
− sin
2
/2
ˆ
∧ (a ∧ ˆ
)
·σ
= (a · ˆ
)( ˆ
·σ ) + cos
a − (a · ˆ
) ˆ
·σ + sin (a ∧ ˆ
)·σ
= R (a)·σ .
6.11.2 Conservation of the Scalar Product under Rotation
with the Definition (6.11)
Solution 6.1 Let us note that we can write, for any (complex) vectors u, a, b, c, d,
…
u = (u · ˆ
) +
u − (u · ˆ
)
= ud + ud ⊥
R(u) = ud + R(ud ⊥ ) = ud + cos ud ⊥ + sin ( ˆ
∧ u)
a ∧ (b ∧ c) =
i jk
klm a j b l c m = (a·c)b − (a·b)c
(a ∧ b) ∧ c = (a·c)b − (c·b)a .
R(d) · R(u) = d ·ud + R(d ⊥ ) · R(u ⊥ ) so