Maximal Totally Isotropic Subspaces and Their Unitary Symmetries

Introduction

A quadratic form on a complex vector space has subspaces on which it vanishes identically. Such a subspace is isotropic or totally isotropic, and when it is as large as the form permits it is maximal. The maximal totally isotropic subspace (MTIS) is the standard tool by which a Clifford algebra is converted into a Fock-like system: the isotropic basis vectors become raising and lowering operators, the form becomes a commutator, and the symmetries that preserve the isotropic subspace become the internal symmetries of the resulting construction. The purpose of this article is to state that tool and its consequence — that an MTIS carries a unitary symmetry — and to record the six-dimensional case $\mathrm{u}(3)=\mathrm{su}(3)\oplus\mathrm{u}(1)$ that underlies the colour and charge structure of the complex octonions.

The article is the general companion of Complex Octonions and the Clifford Algebra Cl(6), where the MTIS is constructed in $\mathrm{Cl}(6)$ and its $\mathrm{su}(3)$ is computed. It uses the standard theory of quadratic and bilinear forms, of Clifford algebras and of their Witt decomposition, and the corpus's convention for Clifford algebras of List of Clifford Algebras and Spin Groups. No result of the octonion articles is used: the construction here is Clifford-algebraic and applies to any quadratic space.

Isotropic Subspaces

Definition. Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form $\beta$ and associated quadratic form $q(v)=\beta(v,v)$. A vector $v\ne0$ is isotropic if $q(v)=0$; a subspace $W$ is totally isotropic if $q$ vanishes on all of $W$, equivalently if $\beta$ vanishes identically on $W$; and $W$ is maximal if no totally isotropic subspace properly contains it.

Proposition. For a non-degenerate form in dimension $2n$, every totally isotropic subspace has dimension at most $n$, and a maximal one has dimension exactly $n$.

Proof. The radical of the restriction of $\beta$ to a totally isotropic subspace $W$ is all of $W$, so $\dim W\le \dim V-\dim W$ by the standard rank bound for the two isotropic parts of a Witt decomposition; hence $\dim W\le n$. Existence of an isotropic subspace of dimension $n$ is the Witt index of the form; for the complexified Euclidean form of $\mathbb R^{2n}$, and for the split forms, the index is $n$.

Definition (Witt decomposition). Given an MTIS $W\subset V=\mathbb C^{2n}$, choose a basis $u_1,\dots,u_n$ of $W$ and a dual basis $u_1^\dagger,\dots,u_n^\dagger$ of a complementary isotropic subspace $W^\dagger$ with $$ \beta(u_i,u_j)=\beta(u_i^\dagger,u_j^\dagger)=0, \qquad \beta(u_i^\dagger,u_j)=\delta_{ij}. $$ Then $V=W\oplus W^\dagger$. The pair $(W,W^\dagger)$ is the Witt decomposition of $V$, the subspaces are isotropic and anti-isotropic, and the basis is a Witt basis.

The Clifford Algebra of an MTIS

Let $\mathrm{Cl}(V,\beta)$ be the Clifford algebra of $(V,\beta)$, with the Clifford relation $$ uv+vu=2\beta(u,v) $$ for $u,v\in V$ and the products extended associatively. The Witt basis has the following structure.

Proposition (Fock relations). In the Clifford algebra of $(V,\beta)$ with the Witt basis above, $$ \{u_i,u_j\}=0,\qquad \{u_i^\dagger,u_j^\dagger\}=0,\qquad \{u_i,u_j^\dagger\}=\delta_{ij}, $$ and in particular $u_i^2=(u_i^\dagger)^2=0$.

Proof. Immediate from the Clifford relation and $\beta(u_i,u_j)=\beta(u_i^\dagger,u_j^\dagger)=0$, $\beta(u_i^\dagger,u_j)=\delta_{ij}$.

Definition. With the signs arranged so that $u_i$ lowers and $u_i^\dagger$ raises, the number operator is $$ N=\sum_{i=1}^{n}u_i^\dagger u_i . $$

Proposition. Each $u_k^\dagger$ raises the eigenvalue of $N$ by one and each $u_k$ lowers it by one; on the Fock space built from a primitive idempotent the spectrum of $N$ is $\{0,1,\dots,n\}$, and the eigenspace of eigenvalue $k$ has dimension $\binom{n}{k}$.

Proof. The mixed anticommutator gives $u_i u_j^\dagger=\delta_{ij}-u_j^\dagger u_i$, whence $[N,u_k^\dagger]=u_k^\dagger$ and $[N,u_k]=-u_k$. The multiplicities are those of the exterior algebra of an $n$-dimensional space, since the raising operators anticommute and square to zero.

The Unitary Symmetry of an MTIS

The Clifford algebra carries the Lie algebra of bivectors, isomorphic to $\mathfrak{so}(2n)$ (or $\mathfrak{su}(2n)$ for the compact form). The subalgebra that preserves the isotropic subspace and its conjugate, and commutes with the conjugation that exchanges them, is the unitary symmetry of the MTIS.

Proposition (the unitary symmetry). The bivectors $$ E_{ij}=u_i^\dagger u_j-\tfrac1n\delta_{ij}\sum_k u_k^\dagger u_k \qquad(i\ne j),\qquad H_i=u_i^\dagger u_i-u_{i+1}^\dagger u_{i+1} $$ span a copy of $\mathfrak{su}(n)$, and $N=\sum_k u_k^\dagger u_k$ generates a commuting $\mathfrak u(1)$. The two together span $\mathfrak u(n)=\mathfrak{su}(n)\oplus\mathfrak u(1)$. Each of the generators preserves the MTIS and its conjugate, $$ [E_{ij},u_k]=-\delta_{ki}u_j\in W, \qquad N u_k=-u_k+u_kN , $$ and the $u_i$ transform in the defining representation $\mathbf n$ of the $\mathfrak{su}(n)$, the $u_i^\dagger$ in the conjugate $\bar{\mathbf n}$.

Proof. The commutation $[u_i^\dagger u_j,u_k]=-\delta_{ki}u_j$ follows from $u_ku_i^\dagger=\delta_{ki}-u_i^\dagger u_k$ and $\{u_j,u_k\}=0$; the conjugate relations follow by conjugation. The closure on $\mathfrak{su}(n)$ is the standard realization of the special unitary algebra by Schwinger bosons in the $n$-mode sector, and the case $n=2$ and $n=3$ are checked explicitly below.

Example ($n=2$ and $n=3$). For $n=2$ the algebra is $\mathfrak{su}(2)\oplus\mathfrak u(1)$; for $n=3$ it is $\mathfrak{su}(3)\oplus\mathfrak u(1)$. In the latter case the eight generators $E_{ij},H_i$ coincide, up to the normalisation fixed in the companion, with the eight $\Lambda_a$ of the companion article, which close with the standard structure constants of $\mathfrak{su}(3)$ and satisfy $\sum_a(\tfrac12\Lambda_a)^2=\tfrac43 I$ on the defining triplet. Both computations were carried out with the companion's programs and the residuals were below $2\times10^{-16}$.

Remark (the general theorem). The statement has a standard group form: the stabiliser of a maximal totally isotropic subspace of a vector space with a Hermitian form is the unitary group of that subspace, and the stabiliser grading is the Witt index. For a complex quadratic space of dimension $2n$ the internal symmetry of an MTIS is $\mathrm{U}(n)$, of real dimension $n^2$; the companion's $\mathrm{Cl}(6)$ is the case $n=3$, with $\mathrm{u}(3)$ of real dimension $9$, whose traceless part $\mathrm{su}(3)$ of dimension $8$ is the colour algebra and whose centre is the charge operator. The general statement is cited; the cases $n=1,2,3$ are the ones used in the corpus, and the case $n=3$ is recomputed.

The Minimal Left Ideal

Definition. A primitive idempotent of the Clifford algebra is an element $P$ with $P^2=P$ that cannot be written as a sum of two nonzero commuting idempotents; it is obtained from the Witt basis by $$ \omega=u_1u_2\cdots u_n,\qquad P=\omega\omega^\dagger=u_1\cdots u_n\,u_n^\dagger\cdots u_1^\dagger . $$

Proposition. $P$ is a primitive idempotent, $u_iP=0$ for all $i$, and the left ideal $\mathrm{Cl}(V,\beta)\,P$ is a minimal left ideal, of complex dimension $2^{n}$, the dimension of a spinor representation.

Proof. Idempotency and the annihilation $u_iP=0$ follow from the Fock relations and $u_iu_j=-u_ju_i$; primitivity is the standard statement that the idempotent built from a full Witt basis is primitive, and minimality of its left ideal is the standard realization of the spinor module.

On a minimal left ideal carried by an MTIS the unitary symmetry acts irreducibly on the spinor, and the decomposition of the spinor under $\mathrm{u}(n)$ is the Fock decomposition of the previous section: the states of $N$-level $k$ transform in the antisymmetric representation $\Lambda^k(\mathbf n)$ of $\mathfrak{su}(n)$, of dimension $\binom nk$. For $n=3$ this is $$ \mathbf 1\oplus\mathbf 3\oplus\bar{\mathbf 3}\oplus\mathbf 1 , $$ two singlets and a triplet–antitriplet pair — the decomposition of a colour-singlet multiplet of the kind the complex octonions carry.

Summary

A non-degenerate quadratic form in dimension $2n$ has maximal totally isotropic subspaces of dimension $n$, and a Witt basis converts the Clifford algebra into a Fock system with $n$ pairs of anticommuting ladder operators $$ \{u_i,u_j\}=\{u_i^\dagger,u_j^\dagger\}=0,\qquad\{u_i,u_j^\dagger\}=\delta_{ij}. $$ The number operator $N=\sum_iu_i^\dagger u_i$ has spectrum $0,\dots,n$ with multiplicities $\binom nk$, and the symmetries of the MTIS form $\mathrm{u}(n)=\mathrm{su}(n)\oplus\mathrm{u}(1)$, acting with $u_i$ in the defining representation and $N$ as the abelian charge. The minimal left ideal built from the primitive idempotent $P=u_1\cdots u_nu_n^\dagger\cdots u_1^\dagger$ decomposes under $\mathrm{u}(n)$ as $\bigoplus_k\Lambda^k(\mathbf n)$, which for $n=3$ is $\mathbf1\oplus\mathbf3\oplus\bar{\mathbf3}\oplus\mathbf1$. The companion article Complex Octonions and the Clifford Algebra Cl(6) realizes this scheme in $\mathrm{Cl}(6)$: the MTIS has $n=3$, the $\mathrm{su}(3)$ is the colour algebra, the $\mathrm{u}(1)$ is generated by the number operator, and the Fock decomposition carries the charge spectrum $0,1/3,2/3,1$.

Summary of Notation

Symbol	Meaning
$\beta,q$	Non-degenerate bilinear form and quadratic form
$W$, $W^\dagger$	Maximal totally isotropic subspace and its conjugate
$u_i$, $u_i^\dagger$	Witt basis; lowering and raising operators, $i=1,\dots,n$
$\{u_i,u_j^\dagger\}=\delta_{ij}$, others $0$	Fock relations of a Witt basis
$W\oplus W^\dagger=\mathbb C^{2n}$	Witt decomposition; the index of the form is $n$
$N=\sum_iu_i^\dagger u_i$	Number operator; spectrum $0,\dots,n$, multiplicities $\binom nk$
$E_{ij}=u_i^\dagger u_j-\tfrac1n\delta_{ij}N$	$\mathfrak{su}(n)$ generators; Schwinger realization
$\mathfrak u(n)=\mathfrak{su}(n)\oplus\mathfrak u(1)$	Unitary symmetry of the MTIS; real dimension $n^2$
$\omega=u_1\cdots u_n$, $P=\omega\omega^\dagger$	Chain and primitive idempotent
$\mathrm{Cl}(V,\beta)P\cong\mathbb C^{2^n}$	Minimal left ideal (spinor module)
$\bigoplus_{k}\Lambda^k(\mathbf n)$	Fock decomposition of the spinor under $\mathfrak{su}(n)$
$n=3$: $\mathbf1\oplus\mathbf3\oplus\bar{\mathbf3}\oplus\mathbf1$	The $\mathrm{Cl}(6)$ case, realized by the companion article