Hermitian Clifford Structures

Introduction

A real Clifford algebra carries the anti-involutions of reversion $\tilde{\ }$, grade involution $\alpha$ and Clifford conjugation $\hat{\ }$, none of which uses a Hermitian form. A Hermitian Clifford structure is a Clifford structure together with a Hermitian form on its module, so that the conjugation of the form, written $x\mapsto x^{\dagger}$, becomes a fourth anti-involution, the Hermitian conjugation, and the Clifford multiplication is required to be compatible with the form in the sense

$$ c(v)^{\dagger}=-c(v) , $$

the multiplication by a real vector being skew-adjoint. This sign is the geometric convention of the corpus, and it makes the Clifford action of a unit vector a unitary operator of the module. The article introduces the Hermitian conjugation on the elements of the algebra, defines the Hermitian Clifford structures on the algebra and on the module, and derives the compatibility statements.

The boundaries. The Clifford algebra, its reversion, grade involution and Clifford conjugation, and the Witt decomposition are Clifford Algebras and Clifford Algebras in Finite Dimensions; the trace form, the dagger and the positivity of the involution are Hilbert Algebras; the Hermitian conjugation of the elements here is the elementary involution of that article read on the Clifford algebra. The Clifford multiplication as an operator, the spinor bundle and the skew-adjointness $c(v)^*=-c(v)$ are The Clifford Multiplication Operator and The Adjoint of the Clifford Multiplication; the adjoints of the operators built from the Hermitian conjugation belong to the - * Operator Theory group of this category and are not touched here. The complex manifolds and the Kähler structures are Complex Manifolds and Kähler Geometry. The base is the complex field $\mathbb{C}$ with a non-degenerate quadratic form; the Hermitian form is positive definite in the definite case and its indefinite variant is named only.

The Hermitian Conjugation on a Complex Clifford Algebra

Definition. Let $\mathbb{C}\ell(V,Q)$ be the complex Clifford algebra of a complex quadratic space with polar form $B$. A Hermitian conjugation is a conjugate-linear anti-involution

$$ x\longmapsto x^{\dagger} , \qquad (x+y)^{\dagger}=x^{\dagger}+y^{\dagger} , \quad (xy)^{\dagger}=y^{\dagger}x^{\dagger} , \quad (x^{\dagger})^{\dagger}=x , \quad (\lambda x)^{\dagger}=\bar\lambda\,x^{\dagger} , $$

together with a non-degenerate Hermitian form $h$ on $\mathbb{C}\ell(V,Q)$ with respect to which $h(x^{\dagger}y,z)=h(y,xz)$; on the vectors it is compatible with the quadratic form.

Proposition. The Hermitian conjugation exists, it is determined up to the unitary group of $h$, and on a vector it satisfies $v^{\dagger}=\bar\cdot$-linear in $v$ with the value determined by the compatibility with $Q$; it commutes with the grade involution, $\alpha(x^{\dagger})=\alpha(x)^{\dagger}$, and it is distinct from the Clifford conjugation $\hat x=\alpha(\tilde x)$ whenever the form is not real.

Proof. The construction is that of a dagger on an algebra with a trace form in Hilbert Algebras: the form $h$ gives the conjugate-linear isomorphism with the dual, and the transpose of the left multiplication supplies the anti-involution; the compatibility with $Q$ fixes the values on the generating vectors up to a unitary; the commutation with $\alpha$ is the compatibility of the grading; the distinction from $\hat x$ is the presence of the complex conjugation of the coefficients.

Remark (the Witt basis). For the complex quadratic space of even dimension $2m$ with the neutral form, a Witt decomposition $\mathbb{C}^{2m}=W\oplus W'$ into maximal isotropic subspaces with a duality pairing supplies a basis $\mathfrak f_1,\dots,\mathfrak f_m$ of $W$ and $\mathfrak f_1^{\dagger},\dots,\mathfrak f_m^{\dagger}$ of $W'$ with

$$ \{\mathfrak f_j,\mathfrak f_k\}=0 , \qquad \{\mathfrak f_j^{\dagger},\mathfrak f_k^{\dagger}\}=0 , \qquad \{\mathfrak f_j,\mathfrak f_k^{\dagger}\}=\delta_{jk} , $$

and the Hermitian conjugation exchanges the two halves of the basis, $\mathfrak f_j^{\dagger}$ being the Hermitian conjugate of $\mathfrak f_j$; the algebra generated by these $2m$ elements is the Hermitian Clifford algebra of the complex quadratic space, and it is the Clifford algebra of the neutral form in a Hermitian presentation.

Proof. The Witt decomposition is the classification of the neutral quadratic forms; the anticommutation relations are imposed for a Clifford system with the pairing $\delta_{jk}$; the exchange by the conjugation is the definition of the dagger on the isotropic generators, and the algebra they generate has dimension $2^{2m}$, hence is the Clifford algebra.

Hermitian Clifford Modules

Definition. A Hermitian Clifford structure on a Clifford module $S$ over $\mathbb{C}\ell(V,Q)$ is a positive definite Hermitian form $\langle\cdot,\cdot\rangle$ on $S$ such that

$$ \langle c(v)s,t\rangle = -\langle s,c(v)t\rangle , \qquad\text{that is}\qquad c(v)^{\dagger}=-c(v) , $$

for every vector $v$ and all $s,t\in S$; the pair $(S,\langle\cdot,\cdot\rangle)$ is a Hermitian Clifford module.

Proposition. On a Hermitian Clifford module the following hold.

(a) The multiplication by a real vector is skew-adjoint, so $c(v)$ is an anti-self-adjoint operator and $ic(v)$ is self-adjoint.

(b) For a unit vector $v$, $q(v)=\pm1$, the operator $c(v)$ is unitary up to the sign of $q$: $c(v)^{\dagger}c(v)=-q(v)\operatorname{id}$.

(c) The Hermitian form is invariant under the Clifford action of the pin and spin groups, in the sense that $h(\rho(x)s,\rho(x)t)=h(s,t)$ for a unit versor $x$ of the appropriate normalisation, and the Clifford action of the group is unitary.

(d) The compatibility is equivalent to $c(v)^{\dagger}=-c(v)$ for all $v$, and it determines the form up to a positive scalar on an irreducible module.

Proof. (a) is the definition and the reality of $q$; (b) $c(v)^{\dagger}c(v)=-c(v)c(v)=-c(v)^2=q(v)$, so the operator has modulus $\lvert q(v)\rvert$; (c) the invariance for the generating reflections is (b), and a product of unitaries is unitary; (d) the equivalence is the definition, and the uniqueness is Schur's lemma on the irreducible module with the two forms proportional and the positivity fixing the scalar.

Remark (the two conventions). The literature of Clifford analysis uses both $c(v)^{\dagger}=-c(v)$ and $c(v)^{\dagger}=c(v)$, the second after replacing $c$ by $ic$ or grading the module by a complex structure; the corpus fixes the geometric convention (a) above, in agreement with the manifold statement $c(v)^*=-c(v)$ of The Adjoint of the Clifford Multiplication, and every display in the category is read with that sign.

The Spinor Inner Product

Definition. A spinor inner product on a Hermitian Clifford module $S$ is a positive definite Hermitian form $\langle\cdot,\cdot\rangle$ as above together with the Clifford action; on the irreducible module of $\mathbb{C}\ell(V,Q)$ it is unique up to a positive scalar by the previous proposition.

Proposition (the spinor inner product and the chirality). In even complex dimension the module splits $S=S^+\oplus S^-$, the chirality grading $\Pi$ is self-adjoint and unitary, $\Pi^{\dagger}=\Pi=\Pi^{-1}$, the two halves are orthogonal, and the Clifford multiplication by a vector interchanges them, $c(v)S^\pm\subseteq S^\mp$, so $c(v)$ is off-diagonal and skew-adjoint.

Proof. The chirality is the product of an orthonormal basis in even dimension, it squares to a scalar that is normalised to $+1$ in the complex case, and it is an involution of the module with the two eigenspaces $S^\pm$; it is unitary because it is a product of unitaries up to the normalisation; the interchange is the anticommutation of $c(v)$ with the product of an odd number of vectors, and skew-adjointness with an off-diagonal matrix is the pairing of the two halves.

Corollary (the Hermitian structure of the spinor bundle). On a spin manifold with a Hermitian Clifford structure the spinor bundle carries the spinor inner product, the Clifford multiplication is skew-adjoint, and the charge conjugation of the spinor bundle is the Hermitian conjugation combined with the chirality; the form is the datum of the real structure of the twistor construction.

Proof. The pointwise statements are the propositions above and the charge-conjugation identification is the standard spinorial reality structure; the last clause is the Hermitian structure used in Twistor Spaces and the Hermitian Structure of a Conformal Manifold.

Worked Cases

The Complex Numbers

For $\mathbb{C}\ell_1$ with $e_1^2=-1$, the algebra is $\mathbb{C}\oplus\mathbb{C}$ as a vector space with the generator $e_1$, and the Hermitian conjugation fixing $i$ with $e_1^{\dagger}=-e_1$ makes $c(e_1)$ skew-adjoint; the module $\mathbb{C}$ carries the standard inner product and $c(e_1)$ is the multiplication by $i$, whose adjoint is the multiplication by $-i$.

The Complexified Plane

For the neutral complex plane with the Witt basis $\mathfrak f,\mathfrak f^{\dagger}$ one has $\{\mathfrak f,\mathfrak f^{\dagger}\}=1$ and the Hermitian Clifford algebra is generated by the two isotropic elements; the module is two-dimensional, $c(\mathfrak f)$ and $c(\mathfrak f^{\dagger})$ are adjoint to within the sign, and the module is the Fock space built from the isotropic generator.

The Quaternions Complexified

For $\mathbb{C}\ell_3\cong M_2(\mathbb{C})$, the Hermitian conjugation is $X\mapsto X^{\dagger}$ on the matrices after the standard identification, the Clifford generators $\sigma_1,\sigma_2,\sigma_3$ are Hermitian matrices with $\sigma_j^{\dagger}=\sigma_j$, and the skew-adjointness $c(v)^{\dagger}=-c(v)$ is the statement that the Clifford generators are $-i$ times the Pauli matrices in the geometric normalisation; the two conventions differ exactly by the factor $i$ named in the remark.

Summary

A Hermitian Clifford structure is a Clifford structure together with a Hermitian form, so that the conjugation of the form, written $x\mapsto x^{\dagger}$, is an anti-involution on the elements with $(xy)^{\dagger}=y^{\dagger}x^{\dagger}$, and the Clifford multiplication is skew-adjoint, $c(v)^{\dagger}=-c(v)$; the multiplication by a unit vector is unitary up to the sign of $q(v)$, and the Clifford action of the versor group is unitary. The Hermitian conjugation is distinct from the Clifford conjugation $\hat x$ of the adjoint theory, and it commutes with the grade involution. A Hermitian Clifford module is a Clifford module with a positive definite form satisfying the skew-adjointness; in even dimension the chirality $\Pi$ is self-adjoint and unitary, the two chiral halves are orthogonal and are interchanged by the multiplication by a vector. The Hermitian Clifford algebra is the algebra of a neutral complex quadratic space in a Witt presentation, generated by isotropic elements $\mathfrak f_j$ and their Hermitian conjugates $\mathfrak f_j^{\dagger}$ with $\{\mathfrak f_j,\mathfrak f_k^{\dagger}\}=\delta_{jk}$; it is the classification of the complex quadratic forms in a Hermitian dress, and it is the algebraic input of the Hermitian Clifford analysis, whose operator theory belongs to Part III. The Clifford multiplication as an operator is The Clifford Multiplication Operator, its metric adjoint is The Adjoint of the Clifford Multiplication, and the form of the spinor bundle is the one used by the twistor construction of Twistor Spaces and the Hermitian Structure of a Conformal Manifold.

Summary of Notation

Symbol	Meaning
$x^{\dagger}$	Hermitian conjugation; anti-involution, conjugate-linear
$h(x^{\dagger}y,z)=h(y,xz)$	The dagger form of the algebra
$\mathfrak f_j,\mathfrak f_j^{\dagger}$	Witt (isotropic) basis; $\{\mathfrak f_j,\mathfrak f_k^{\dagger}\}=\delta_{jk}$
$c(v)^{\dagger}=-c(v)$	Skew-adjointness; geometric convention
$\Pi^{\dagger}=\Pi=\Pi^{-1}$	Self-adjoint unitary chirality in even dimension
$c(v)S^\pm\subseteq S^\mp$	The multiplication by a vector interchanges the chiral halves
$\hat x=\alpha(\tilde x)$	Clifford conjugation; not the Hermitian conjugation