Example. Real structures in chosen complex coordinates [fcap-003W]
AGENTDRAFTED
Why does the complex Clifford algebra forget a real signature, while
its real vectors still remember it? Let \(V\) be a finite-dimensional real
space with nondegenerate quadratic form \(Q\). Its complexification
is the scalar extension
\(V_{\mathbb C}=\mathbb C\otimes _{\mathbb R}V\). Complex conjugation on
the first factor fixes \(V\) and induces the conjugate-linear algebra
involution
\[\sigma (z\otimes a)=\overline z\otimes a\]
on \(\mathbb C\otimes _{\mathbb R}\mathcal {C}\kern -2pt\ell _{\mathbb R}(Q)\).
It preserves product order and conjugates scalar coefficients; the
grade reversion instead reverses product order and is
complex-linear after scalar extension.
A real structure on a complex vector space or algebra means a
map with \(\sigma (\lambda v+\mu w)=\overline \lambda \sigma (v)+
\overline \mu \sigma (w)\) and \(\sigma ^2=1\), an algebra automorphism in the
algebra case.
The fixed algebra here is \(1\otimes \mathcal {C}\kern -2pt\ell _{\mathbb R}(Q)\): write an element
uniquely as \(1\otimes a+i\otimes b\) with \(a,b\) real to check this.
For the underlying real module of \(\mathcal {C}\kern -2pt\ell _{\mathbb R}(Q)\), this
coefficient map is TauCeti.tmulConj.
Its involution law is TauCeti.tmulConj_involutive, and
its fixed vectors are described by
TauCeti.tmulConj_eq_self_iff.
These supporting declarations concern the conjugate-linear involution and
fixed vectors; multiplicativity and the coordinate comparisons below are
the algebra calculations given here.
This is the conjugation construction in
[meinrenken2013clifford, Section 2.2.6, p. 34].
Meinrenken's symmetric form is \(B=B_Q/2\), so his generator relation is
\(uv+vu=2B(u,v)=B_Q(u,v)\). The following coordinate calculation applies
that construction to the scalar-extension isomorphism of Theorem [fcap-003V].
Choose an orthogonal real basis \(e_1,\ldots ,e_n\) with
\(Q(e_j)=1\) for \(j\le p\) and \(Q(e_j)=-1\) for \(j>p\), where \(n=p+q\).
These are the signature signs of Convention [fcap-0011].
On \(\mathbb C^n\), put \(Q_0(z)=\sum _j z_j^2\), with coordinate vectors
\(u_j\). The chosen complex isometry
\[T(e_j)=\begin {cases}u_j&j\le p,\\i u_j&j>p\end {cases}\]
induces \(\mathcal {C}\kern -2pt\ell (T):\mathcal {C}\kern -2pt\ell _{\mathbb C}(Q_{\mathbb C})
\simeq \mathcal {C}\kern -2pt\ell _{\mathbb C}(Q_0)\). Transporting \(\sigma \) through this map and
Theorem [fcap-003V] gives \(\sigma _T\) on the standard complex algebra. On its
generators,
\[\sigma _T(\iota (u_j))=\begin {cases}
\iota (u_j)&j\le p,\\-\iota (u_j)&j>p.
\end {cases}\]
Indeed, \(T^{-1}(u_j)=-i e_j\) for a negative direction, and conjugation
sends this to \(i e_j\), whose image under \(T\) is \(-u_j\).
The fixed real generators are consequently \(\iota (u_j)\) and
\(i\iota (u_j)\) in the positive and negative directions, respectively.
Already in dimension one, the same complex algebra
\(A=\mathbb C[u]/(u^2-1)\) has two different real structures:
\[\begin {aligned}
\sigma _+(z+wu)&=\overline z+\overline w u,
& A^{\sigma _+}&=\mathbb R[ u]\simeq \mathbb R\oplus \mathbb R,\\
\sigma _-(z+wu)&=\overline z-\overline w u,
& A^{\sigma _-}&=\mathbb R[iu]\simeq \mathbb C.
\end {aligned}\]
The first real-algebra isomorphism evaluates at \(u=1,-1\); the second uses
\((iu)^2=-1\). Thus forgetting the real structure loses the distinction
between \(\mathcal {C}\kern -2pt\ell (1,0)\) and \(\mathcal {C}\kern -2pt\ell (0,1)\). If a different complex isometry is
\(T'=UT\), then
\(\sigma _{T'}=\mathcal {C}\kern -2pt\ell (U)\sigma _T\mathcal {C}\kern -2pt\ell (U)^{-1}\). The real structure moves with
the coordinates; it is not generally entrywise conjugation in every model.
The real-algebra identifications in this example are formalized as
TauCeti.realCliffordOneZeroEquivProd and
TauCeti.realCliffordZeroOneEquivComplex.
These declarations identify the two real Clifford algebras; the comparison
of their conjugations inside \(A\) is the calculation above.
Choosing the polarization data for a spinor
carrier is a further choice. For \(Q(x,y)=x^2+y^2\), the complex isotropic
lines
\[W=\mathbb C(e_1+i e_2),\qquad
W'=\mathbb C(e_1-i e_2)\]
are complementary, and conjugation exchanges them. Their polar pairing is
nonzero, so they give the split model after normalizing a dual basis.
The carrier \(S=\bigwedge W\) uses \(W\) for creation and \(W'\) for
contraction, as in Definition [fcap-001I]. Conjugation therefore maps this exterior
space to \(\bigwedge W'\), rather than supplying a conjugation on \(S\)
compatible with the same chosen operators. Comparing these carriers needs
an additional intertwining map. This explains why the canonical real
structure on the Clifford algebra alone does not choose a real structure
on a particular spinor module. The creation/contraction construction is
[chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44].
Definition [ca-000J]