Darboux-Bäcklund Transformations for Spin-Valued Linear Problems
39
Clifford algebra (denoted by C(V , Q) or C(p, q), where (p, q) is the signature
of Q) is generated by V using linear operations and the Clifford product, see, e.g.,
[18, 21].
Let e 1 , . . . , e N be an orthonormal basis in V . In other words, see (2), we have
e
2
j = ±1 ,
e j e k = −e k e j .
(4)
Then, dim C(V , Q) = 2 N and the Clifford algebra is spanned by 1 (scalars), e k
(vectors), e j e k (j < k) (bivectors) , multivectors, and e 1 e 2 . . . e N (pseudoscalars).
Well-known examples: Pauli matrices (e k = σ k , N = 3, p = 3, q = 0) and
Dirac matrices (e k = γ k , N = 4, p = 1, q = 3).
The vector space V , generating the Clifford algebra C(p, q) (p + q = N ), can
be identified with the pseudo-Euclidean space spanned by e 1 , e 2 . . . , e N . Reflection
with respect to the hyperplane orthogonal to an invertible n ∈ V can be represented
as
v
= −nvn
−1 ,
hence v
= −(2 | v − vn)n
−1
= v −
2 | v
n | n
n .
(5)
Indeed, if v = v n + v t (where v n is normal and v t is parallel to the hyperplane), then
v = v t − v n . Thus: −n(v t + v n )n −1 = v t nn −1 − v n nn −1 = v .
By the Cartan–Dieudonné theorem any isometry of V can be represented as a
composition of at most N, say k, reflections. Therefore, using (5), we have
v
= (−1)
k n k n k−1 . . . n 1 v n
−1
1 n
−1
2 . . . n
−1
k .
(6)
Hence, it is natural to consider multiplicative groups in Clifford algebras:
• Lipschitz group Γ (V , Q): products of invertible vectors,
• Pin group
Pin(V , Q): products of unit vectors,
• Spin group
Spin(V , Q): products of even number of unit vectors
(often we omit Q, writing Spin(V ), etc.). Thus Eq. (6) can be rewritten as
C(p, q) ⊃ V v → v
= Ψ v Ψ
−1
∈ V ,
(7)
where Ψ ∈ Pin(p, q), which is equivalent to an orthogonal transformation in V :
R
p,q
⊃ V v → v
= R v ∈ V
(8)
where R ∈ O(p, q). Obviously, −Ψ and Ψ yield the same isometry R. Therefore,
we have double coverings, like Pin(N) → O(N) or Spin(N) → SO(N).
The Lie algebra spin(p, q) of the Spin group Spin(p, q) is spanned by bivectors
e j e k (1 ≤ j < k ≤ N ). Note that [e m e k , e j e k ] = 2 k | e k e j e m (m = j = k = m).
39
Clifford algebra (denoted by C(V , Q) or C(p, q), where (p, q) is the signature
of Q) is generated by V using linear operations and the Clifford product, see, e.g.,
[18, 21].
Let e 1 , . . . , e N be an orthonormal basis in V . In other words, see (2), we have
e
2
j = ±1 ,
e j e k = −e k e j .
(4)
Then, dim C(V , Q) = 2 N and the Clifford algebra is spanned by 1 (scalars), e k
(vectors), e j e k (j < k) (bivectors) , multivectors, and e 1 e 2 . . . e N (pseudoscalars).
Well-known examples: Pauli matrices (e k = σ k , N = 3, p = 3, q = 0) and
Dirac matrices (e k = γ k , N = 4, p = 1, q = 3).
The vector space V , generating the Clifford algebra C(p, q) (p + q = N ), can
be identified with the pseudo-Euclidean space spanned by e 1 , e 2 . . . , e N . Reflection
with respect to the hyperplane orthogonal to an invertible n ∈ V can be represented
as
v
= −nvn
−1 ,
hence v
= −(2 | v − vn)n
−1
= v −
2 | v
n | n
n .
(5)
Indeed, if v = v n + v t (where v n is normal and v t is parallel to the hyperplane), then
v = v t − v n . Thus: −n(v t + v n )n −1 = v t nn −1 − v n nn −1 = v .
By the Cartan–Dieudonné theorem any isometry of V can be represented as a
composition of at most N, say k, reflections. Therefore, using (5), we have
v
= (−1)
k n k n k−1 . . . n 1 v n
−1
1 n
−1
2 . . . n
−1
k .
(6)
Hence, it is natural to consider multiplicative groups in Clifford algebras:
• Lipschitz group Γ (V , Q): products of invertible vectors,
• Pin group
Pin(V , Q): products of unit vectors,
• Spin group
Spin(V , Q): products of even number of unit vectors
(often we omit Q, writing Spin(V ), etc.). Thus Eq. (6) can be rewritten as
C(p, q) ⊃ V v → v
= Ψ v Ψ
−1
∈ V ,
(7)
where Ψ ∈ Pin(p, q), which is equivalent to an orthogonal transformation in V :
R
p,q
⊃ V v → v
= R v ∈ V
(8)
where R ∈ O(p, q). Obviously, −Ψ and Ψ yield the same isometry R. Therefore,
we have double coverings, like Pin(N) → O(N) or Spin(N) → SO(N).
The Lie algebra spin(p, q) of the Spin group Spin(p, q) is spanned by bivectors
e j e k (1 ≤ j < k ≤ N ). Note that [e m e k , e j e k ] = 2 k | e k e j e m (m = j = k = m).