The Krein Level Sets and the Hyperbolic Structure

Introduction

The Krein form is indefinite, so its nonzero level sets are neither spheres nor compact: they are hyperboloids, and the geometry they carry is hyperbolic rather than elliptic. This article describes the three sign level sets of the form $[\tilde{Q},\tilde{Q}]=\sum_{\mu}\varepsilon_{\mu}\bar Q_{\mu}Q'_{\mu}$ — the positive set $\{[\tilde{Q},\tilde{Q}]=1\}$, the negative set $\{[\tilde{Q},\tilde{Q}]=-1\}$ and the null set $\{[\tilde{Q},\tilde{Q}]=0\}$ — and the two models of hyperbolic space that the sign sets produce: the complex hyperbolic space of positive definite complex lines, which is the open unit ball of $\mathbb{V}_{\mathbb{B}}\cong\mathbb{C}^{3}$ and coincides with the symmetric space $U(1,3)/(U(1)\times U(3))$ of The Krein Cartan Decomposition of the Operator Algebra, and the real hyperbolic space $H^{3}$ of the Minkowski slice, which is the sheet of the real hyperboloid. The isotropic structure at the boundary is The Isotropic Structure of the Krein Form; the cone of the form is Indefinite Positivity and the Krein Cone of the Biquaternion Algebra.

Conventions. $e_0=1$, $e_k^{2}=-e_0$, central scalar imaginary $i$, $\mathrm{Sc}$ the scalar part; the Krein form is $[\tilde{Q},\tilde{Q}']=\sum_{\mu}\varepsilon_{\mu}\bar Q_{\mu}Q'_{\mu}$ with $\varepsilon=(1,-1,-1,-1)$; the Hermitian form is $\langle\tilde{Q},\tilde{Q}'\rangle=\sum_{\mu}\bar Q_{\mu}Q'_{\mu}$ with $\|\tilde{Q}\|_E^{2}=\langle\tilde{Q},\tilde{Q}\rangle$.

The Three Level Sets

Theorem (the level sets and their homotopy types). In the splitting $\tilde{Q}=c+v$ into the centre and vector parts,

$$ \{\tilde{Q}:[\tilde{Q},\tilde{Q}]=1\}=\Bigl\{(c,v):\|c\|_E^{2}=1+\|v\|_E^{2}\Bigr\}\cong S^{1}\times\mathbb{R}^{6}, $$ $$ \{\tilde{Q}:[\tilde{Q},\tilde{Q}]=-1\}=\Bigl\{(c,v):\|v\|_E^{2}=1+\|c\|_E^{2}\Bigr\}\cong S^{5}\times\mathbb{R}^{2}, $$ $$ \{\tilde{Q}:[\tilde{Q},\tilde{Q}]=0\}=\{(c,v):\|c\|_E=\|v\|_E\}\cong S^{1}\times S^{5}\times\mathbb{R}_{>0}, $$

all three of real dimension $7$; the first is homotopy equivalent to $S^{1}$, the second to $S^{5}$, the third to $S^{1}\times S^{5}$.

Proof. For the first, a pair $(c,v)$ satisfies the equation exactly when $c$ lies on the circle of radius $\sqrt{1+\|v\|_E^{2}}$ in the centre, so the map $(c,v)\mapsto(\|v\|_E/c\cdot c,\ v)$ identifies the set with $S^{1}\times\mathbb{R}^{6}$: the pair $(v,c/\|c\|_E)$ is free and determines $c$. For the second, in the same way $\|v\|_E=\sqrt{1+\|c\|_E^{2}}>0$, so $v$ is determined by its direction in the five-sphere and by $c$, which is free in $\mathbb{C}$; the set is $S^{5}\times\mathbb{R}^{2}$. For the third, both norms are equal to some $t>0$, and the set is $S^{1}\times S^{5}\times\mathbb{R}_{>0}$ by the two directions and the radius.

Remark (what is lost and what is kept). Compared with the Euclidean sphere, the positive level set loses compactness and the sphere: it is the product of a circle with a seven-dimensional Euclidean space, a hyperboloid of revolution. All three sets are connected and non-compact, as the signature $(2,6)$ over $\mathbb{R}$ requires, and the negative set has the homotopy type of $S^{5}$ alone because its vector part is forced away from the origin while its scalar part is free.

The Positive Region and the Ball of Positive Lines

Definition. The positive region is the open cone $\mathcal{P}=\{\tilde{Q}:[\tilde{Q},\tilde{Q}]>0\}$, the interior of the Krein cone; the negative region is $\{\tilde{Q}:[\tilde{Q},\tilde{Q}]<0\}$; and a positive line is a complex line consisting of positive elements.

Theorem (the positive region retracts onto a circle). The positive region is $\{\|c\|_E>\|v\|_E\}$, it is a cone, and it is homotopy equivalent to $S^{1}$; the negative region is homotopy equivalent to $S^{5}$.

Proof. For the first, the map $\tilde{Q}\mapsto c/\|c\|_E$ is defined on the positive region, since $\|c\|_E>\|v\|_E\ge0$ forces $c\neq0$; for fixed $c$ the vector part ranges over the open ball of radius $\|c\|_E$, a contractible set, so the projection onto $\mathbb{C}\setminus\{0\}\simeq S^{1}$ is a homotopy equivalence. For the second, $\|v\|_E>\|c\|_E$ forces $v\neq0$, and the projection onto $v/\|v\|_E\in S^{5}$ has contractible fibres again.

Theorem (the positive lines form the complex hyperbolic ball). The positive lines are exactly the lines $\mathbb{C}(e_0+\tilde{V})$ with $\|\tilde{V}\|_E<1$; the assignment $\tilde{V}\mapsto\mathbb{C}(e_0+\tilde{V})$ is a bijection of the open unit ball of $\mathbb{V}_{\mathbb{B}}\cong\mathbb{C}^{3}$ onto the set of positive lines. That set is therefore a real manifold of dimension $6$, isomorphic as a homogeneous space to

$$ U(1,3)/(U(1)\times U(3)), $$

the complex hyperbolic space $\mathbb{CH}^{3}$, and the isometry group $U(1,3)$ of the Krein form acts on it transitively with the stabiliser $U(1)\times U(3)$ of a positive line.

Proof. A positive line is not contained in $\mathbb{V}_{\mathbb{B}}$, since the form is negative definite there; choosing the unique representative $e_0+\tilde{V}$ and using $[e_0+\tilde{V},e_0+\tilde{V}]=1-\|\tilde{V}\|_E^{2}$ shows that positivity is exactly $\|\tilde{V}\|_E<1$, and the assignment is bijective. The stabiliser of the line $\mathbb{C}e_0$ in $U(1,3)$ consists of the isometries preserving the canonical fundamental decomposition, which is $U(1)\times U(3)$ (The Krein Isometry Group and Its $J$-Contractions), whence the homogeneous description; the two realisations agree because both are the symmetric space of the same Cartan pair (The Krein Cartan Decomposition of the Operator Algebra).

Remark (the boundary). By The Isotropic Structure of the Krein Form the closure of the ball adds the isotropic lines, $\|\tilde{V}\|_E=1$, a copy of $S^{5}$: it is the boundary sphere of the complex hyperbolic ball, of real dimension $5$, while the ball itself has real dimension $6$.

The Minkowski Slices

Theorem (the real hyperboloids). On the Hermitian subspace $\mathbb{M}_{+}$ and on the real quaternion subspace $\mathbb{H}_{\mathbb{B}}$ the Krein form is the interval form of Minkowski space, and with the real coordinates $q$,

$$ [\tilde{Q},\tilde{Q}]=q_0^{2}-q_1^{2}-q_2^{2}-q_3^{2}. $$

The positive level set is the two-sheeted hyperboloid, whose sheet $q_0>0$ is given by $q_0=\sqrt{1+\|\mathbf{q}\|^{2}}$ with $\mathbf{q}\in\mathbb{R}^{3}$ free, hence is $\mathbb{R}^{3}$; the negative level set is the one-sheeted hyperboloid $q_0^{2}=\|\mathbf{q}\|^{2}-1$, diffeomorphic to $S^{2}\times\mathbb{R}$; and the null set is the light cone, of two nappes.

Proof. The restricted form is $\sum_{\mu}\varepsilon_{\mu}q_{\mu}^{2}$ in the four real coordinates of the slice (The Krein Gram Matrix and the Restrictions of the Form). Solving $q_0^{2}-\|\mathbf{q}\|^{2}=1$ for $q_0$ gives the two sheets, each parametrised by $\mathbf{q}\in\mathbb{R}^{3}$. Solving $q_0^{2}-\|\mathbf{q}\|^{2}=-1$ gives $\|\mathbf{q}\|\ge1$ with $q_0$ free, so the set is $\mathbb{R}\times S^{2}$ by the direction of $\mathbf{q}$. The null equation factors as $(q_0-\|\mathbf{q}\|)(q_0+\|\mathbf{q}\|)=0$, giving the two nappes of the light cone.

Corollary (the hyperboloid model of $H^{3}$). The sheet $q_0=\sqrt{1+\|\mathbf{q}\|^{2}} $ carries the Riemannian metric induced by the Lorentz form and is the hyperboloid model of the real hyperbolic space $H^{3}$; the Lorentz boosts of Biquaternion Rotations and Lorentz Transformations act on it by its isometries. The complex hyperbolic ball of the preceeding section is the complexification of this picture: dimension $6$ instead of $3$, the sphere $S^{5}$ instead of $S^{2}$, and the group $U(1,3)$ instead of the Lorentz group.

The Level Sets Compared

Remark. The last row belongs to the definite Hermitian form and is quoted for contrast: it is the sphere of The Euclidean Topology of the Biquaternion Algebra, the only compact member of the table, and the reason the indefinite level sets carry hyperbolic rather than elliptic geometry.

Worked Examples

A point of the positive level set. $\tilde{Q}=e_0$: $[\tilde{Q},\tilde{Q}]=1$, on the positive set, the base point of the ball.

A point of the negative level set. $\tilde{Q}=e_1+e_2$: $[\tilde{Q},\tilde{Q}]=-2$, so $\tilde{Q}/\sqrt2$ has square $-1$.

Two points of the null set. $e_0+e_1$ and $(e_0+e_1)/\sqrt2$: the first has square $0$ and norm $2$, the second square $0$ and Euclidean norm $1$; the second is on the link, and both are on the boundary of the ball.

A line inside the ball. $\mathbb{C}(e_0+0.5e_1)$: $\|\tilde{V}\|_E=0.5<1$, a positive line, a point of the complex hyperbolic ball.

A line on the boundary. $\mathbb{C}(e_0+e_1)$: $\|\tilde{V}\|_E=1$, an isotropic line, a point of the boundary sphere $S^{5}$.

A real timelike line. $\tilde{Q}=e_0+0.5e_1$ in $\mathbb{H}_{\mathbb{B}}$: $[\tilde{Q},\tilde{Q}]=0.75>0$, a point of the Minkowski hyperboloid sheet.

Summary

The sign level sets of the Krein form are the hyperboloids $\{[\tilde{Q},\tilde{Q}]=1\}\cong S^{1}\times\mathbb{R}^{6}\simeq S^{1}$ and $\{[\tilde{Q},\tilde{Q}]=-1\}\cong S^{5}\times\mathbb{R}^{2}\simeq S^{5}$, and the null set $S^{1}\times S^{5}\times\mathbb{R}_{>0}\simeq S^{1}\times S^{5}$; all are non-compact of real dimension $7$, in contrast with the compact Euclidean sphere $S^{7}$. The positive region $\{\|c\|_E>\|v\|_E\}$ retracts onto the circle of phases and the negative region onto $S^{5}$. The positive lines are the lines $\mathbb{C}(e_0+\tilde{V})$ with $\|\tilde{V}\|_E<1$, so they form the open unit ball of $\mathbb{C}^{3}$, of real dimension $6$, which is the complex hyperbolic space $\mathbb{CH}^{3}=U(1,3)/(U(1)\times U(3))$ with boundary the isotropic sphere $S^{5}$. On a Minkowski slice the same picture degenerates to the sheet $q_0=\sqrt{1+\|\mathbf{q}\|^{2}}$ of the real hyperboloid, the hyperboloid model of the real hyperbolic space $H^{3}$, the one-sheeted hyperboloid $S^{2}\times\mathbb{R}$, and the two nappes of the light cone.

Summary of Notation

Further Reading

• The Isotropic Structure of the Krein Form (articles_maths/the-isotropic-structure-of-the-krein-form.md), for the boundary sphere and the null cone
• Krein Orthogonality and the Fundamental Decomposition (articles_maths/krein-orthogonality-and-the-fundamental-decomposition.md), for the parametrisation of the maximal positive definite subspaces
• Indefinite Positivity and the Krein Cone of the Biquaternion Algebra (articles_maths/indefinite-positivity-and-the-krein-cone-of-the-biquaternion-algebra.md), for the cone and its interior
• The Krein Cartan Decomposition of the Operator Algebra (articles_maths/the-krein-cartan-decomposition-of-the-operator-algebra.md), for the symmetric space $U(1,3)/(U(1)\times U(3))$
• Biquaternion Rotations and Lorentz Transformations (articles_maths/biquaternion-rotations-and-lorentz-transformations.md), for the boosts acting on the Minkowski hyperboloid
• The Euclidean Topology of the Biquaternion Algebra (articles_maths/the-euclidean-topology-of-the-biquaternion-algebra.md), for the compact sphere used here for contrast