The Polar Decomposition and Its Three Pairings
Introduction
Every biquaternion off the null cone is the ordered product of four factors: a positive scale, a central phase, a Hermitian boost and a real quaternion rotor,
$$ \boxed{\;\tilde Q=r\,\exp(i\alpha)\,\tilde U_{\mathrm b}\,\tilde u_{\mathrm r} =r\,\exp(i\alpha)\,\exp\Bigl(i\tfrac{\psi_w}{2}\vec u_{\mathrm b}\Bigr)\exp\Bigl(\tfrac{\theta}{2}\vec u_{\mathrm r}\Bigr)\;}. $$
The four factors carry $1+1+3+3=8$ real parameters, the real dimension of the algebra. The word is a change of coordinates on the complement of the zero-divisor cone, not a dynamical decomposition: it says what an element is at one instant and not how it evolves.
The article is the hub of the polar programme. It fixes the word, brackets it in the three ways the corpus uses, compares the bracketings, and records what each factor does to the remarkable subspaces. The detail now lives in three synthesis articles: The Three Polar Factors compares the three first factors, The Cartan Pairing, the Amplitude and the Operator-to-State Map develops the amplitude–phase reading, and The Hamilton Pairing: Mass as Modulus and the Complex Angle develops the invariant and the complex angle. The hub states the grammar once and links.
Two distinctions organise the article and are worth naming at the start. A pairing is a bracketing of the four-factor product into two factors; a split is a direct-sum decomposition of the algebra as a vector space. The corpus has three of each, and the two threes are not the same three. §The Three Splits separates them.
The Results It Uses
| Mathematical Content | Source Mathematical Article |
|---|---|
| The four factors, their ranges, their dimensions, the theorem and the uniqueness | The Polar Decomposition of Biquaternions |
| The exponential word, the logarithm, the kernel and the boundary limits | Exponential Polar Decomposition of Biquaternions |
| The quaternion pairing, its construction and its domain | The Quaternion Pairing of the Polar Decomposition of Biquaternions |
| The Hamilton pairing, the complex modulus and the complex angle | The Hamilton Root Pairing of the Polar Decomposition of Biquaternions |
| The Cartan pairing and the matrix polar decomposition | The Cartan Hermitian Pairing of the Polar Decomposition of Biquaternions |
| The generalized pairing on the null cone | The Generalized Polar Decomposition of a Biquaternion |
| The action of the four factors on the sectors and the six remarkable subspaces | The Polar Decomposition and Remarkable Subspaces |
The Four Factors
| # | Factor | Name | Exponent | Root | Range | Real dimension | Action on the algebra |
|---|---|---|---|---|---|---|---|
| 1 | $r$ | scale | — | — | $r>0$ | $1$ | dilation by $r^2$ |
| 2 | $\exp(i\alpha)$ | central phase | $i\alpha$ | $i$ | $\alpha\in(-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ | $1$ | multiplication by the scalar |
| 3 | $\tilde U_{\mathrm b}=\exp(i\tfrac{\psi_w}{2}\vec u_{\mathrm b})$ | boost | $i\tfrac{\psi_w}{2}\vec u_{\mathrm b}$ | $i\vec u_{\mathrm b}$ | $\psi_w\ge0$, $\vec u_{\mathrm b}\in\mathcal{A}$ | $3$ | congruence $X\mapsto \tilde U_{\mathrm b}X\tilde U_{\mathrm b}^{*}$ |
| 4 | $\tilde u_{\mathrm r}=\exp(\tfrac{\theta}{2}\vec u_{\mathrm r})$ | rotor | $\tfrac{\theta}{2}\vec u_{\mathrm r}$ | $\vec u_{\mathrm r}$ | $\theta\in[0,2\pi]$, $\vec u_{\mathrm r}\in\mathcal{A}$ | $3$ | conjugation $X\mapsto \tilde u_{\mathrm r}X\tilde u_{\mathrm r}^{-1}$ |
The axis sphere $\mathcal{A}$ is the set of unit real pure quaternions, the roots of $-e_t$. The phase is the only factor whose action is not a sandwich; its action is scalar multiplication.
The scale and the phase are central, the boost is Hermitian positive and non-compact, and the rotor is compact and real. The order of the factors is forced: the scale and the phase commute with everything, the boost stands on the left of the rotor, and the rotor is what remains after the Hermitian part is divided out. The word is unique once the branch of the square root is fixed.
Reading the Eight Numbers
The word returns the physical description of an element in eight real numbers: the scale $r$, the phase $\alpha$, the rapidity $\psi_w$ and the two angles of the boost axis $\vec u_{\mathrm b}$, and the rotation angle $\theta$ and the two angles of the rotor axis $\vec u_{\mathrm r}$.
The first two are the central data, the next three the non-compact data, the last three the compact data. The scale and the phase are fixed by the norm $N$: $r=\lvert N\rvert^{1/2}$ and $\alpha=\tfrac12\arg N$. The word's rapidity $\psi_w$ is fixed by the ratio of the two invariants,
$$ \cosh\psi_w=\frac{\langle\tilde Q,\tilde Q\rangle_{*}}{\lvert N\rvert}, $$
and the two axes are fixed by the phases of the four complex coordinates. The pair $(N,\langle\tilde Q,\tilde Q\rangle_{*})$ fixes the scale, the phase and the rapidity and cannot fix the two directions: a rotation of the vector part leaves both invariants unchanged and turns the axes.
The word is a parametrization of the eight-dimensional algebra and not a constraint on it. It is surjective onto the units and not injective where an axis degenerates, at $\psi_w=0$ and at $\theta=0$.
The Domain
The word exists if and only if $N\neq0$, that is, if and only if the element is outside the zero-divisor cone. On the cone the product of the four factors has norm of modulus $r^{2}>0$, so no factorisation exists. The condition is one complex equation, hence two real equations, and the cone has real codimension two: the domain is the complement of a six-real-dimensional cone in the eight-real-dimensional algebra.
The failure is physical. A lightlike four-vector has no rest frame, so the scale degenerates and there is no boost split: there is no frame in which its spatial part vanishes and no rapidity that carries it to that frame. The algorithm stops on the cone for the same reason a boost parametrisation stops at the speed of light.
The three pairings have smaller domains than the word, and the table states each once. The distinction between the factorisation and the construction from the parts is owned by Conventions in the Biquaternion Universe, §The Domains of the Three Pairings; it is restated here because every statement below must be read against it.
| pairing | factorisation exists | construction exists | witness of the smaller domain |
|---|---|---|---|
| word | $N\neq0$ | same | any element of the cone |
| quaternion | $N\neq0$ | $N\neq0$ and $\tilde q_r\neq0$ | $i\vec e_x+i\vec e_y$: $N=-2$, $\tilde q_r=0$ |
| Hamilton | $N\neq0$ and $\langle\vec Q,\vec Q\rangle_{\natural}\neq0$ | same | $e_t+\vec e_x+i\vec e_y$: $N=1$, $B=0$ |
| Cartan | $N\neq0$; generalized: all $\tilde Q$ | same | $e_t+i\vec e_x$: on the cone |
The domains are not nested. The quaternion construction fails at $\tilde q_r=0$; the Hamilton pairing fails when the vector part is null; the Cartan pairing is the widest and the generalized pairing extends it to the cone with a partial isometry. A statement proved in one pairing does not transport to another without checking the witness.
The Three Pairings
The four-factor word is the reference. The three pairings are its three bracketings into two factors, and the table records which pair of factors each one groups together.
| Pairing | First factor | Second factor | What it separates |
|---|---|---|---|
| Quaternion | $\tilde q=r\tilde u_{\mathrm r}$ | $\exp(i\tilde\Psi)$ | scale and rotor / phase and boost |
| Hamilton | $\Rho=r\exp(i\alpha)$ | $\exp(\vec U\Theta)$ | scale and phase / boost and rotor |
| Cartan | $\tilde H=r\tilde U_{\mathrm b}$ | $\tilde U=\exp(i\alpha)\tilde u_{\mathrm r}$ | scale and boost / phase and rotor |
The Quaternion Pairing
The quaternion pairing writes the element as a real quaternion modulus times the exponential of a real quaternion angle:
$$ \boxed{\;\tilde Q=\tilde q\,\exp(i\tilde\Psi), \qquad \tilde q=r\,\tilde u_{\mathrm r}, \qquad \exp(i\tilde\Psi)=\exp(i\alpha)\,\tilde u_{\mathrm r}^{-1}\tilde U_{\mathrm b}\tilde u_{\mathrm r}\;}. $$
The modulus is an ordinary quaternion in its polar form, and it is the spatial frame. The exponent has scalar part $\alpha$ and pure part $\tfrac{\psi_w}{2}\vec u_{\mathrm b,2}$, with the boost axis rotated by the rotor. The reading is developed in The Three Polar Factors, §The Quaternion Factor.
The Hamilton Pairing
The Hamilton pairing writes the element as a modulus times a complex cosine plus a complex axis times the complex sine:
$$ \boxed{\;\tilde Q=\Rho\bigl(\cos\Theta+\vec\Xi\sin\Theta\bigr), \qquad \Rho=\sqrt N, \qquad \vec\Xi=\frac{\vec Q}{B}, \qquad B=\sqrt{\langle\vec Q,\vec Q\rangle_{\natural}}\;}. $$
The modulus is the only central scalar of the three pairings. The axis and the angle carry the boost and the rotor; on aligned axes the real part of $\Theta$ is half the rotation and the imaginary part is half the word's rapidity $\psi_w$, and in general the two do not separate. The physical reading — the mass for a four-momentum, the amplitude otherwise — is developed in The Hamilton Pairing, §The Central Modulus.
The Cartan Pairing
The Cartan pairing writes the element as a Hermitian positive factor times a unitary factor:
$$ \boxed{\;\tilde Q=\tilde H\tilde U, \qquad \tilde H=\sqrt{\tilde Q\tilde Q^{*}}=r\tilde U_{\mathrm b}, \qquad \tilde U=\tilde H^{-1}\tilde Q=\exp(i\alpha)\tilde u_{\mathrm r}\;}. $$
The first factor is the amplitude, the part that changes norms and not phases; the second is the phase, the part that preserves the inner product. The state reading is developed in The Cartan Pairing, the Amplitude and the Operator-to-State Map.
The Cartan pairing has the widest construction domain, the quaternion pairing the most direct electromagnetic reading, and the Hamilton pairing the most direct Lorentz-group reading. None is more fundamental than the others, and none adds information to the four-factor word.
The Three Splits
A split is a direct-sum decomposition of the algebra, not a bracketing of a product. The algebra has three natural two-way splits, and each belongs to a different reading.
| split | pieces | used by | physical reading |
|---|---|---|---|
| real vs. imaginary coefficients | $\mathbb{H}_{\mathbb{B}}\oplus i\mathbb{H}_{\mathbb{B}}$ | the quaternion construction | the real quaternion frame and the imaginary-coefficient part |
| Hermitian vs. anti-Hermitian | $\mathbb{M}_+\oplus\mathbb{M}_-$ | the physical sectors; the Cartan exponents | the informational and material generators |
| scalar vs. vector | $\mathbb{C}_{\mathbb{B}}\oplus\operatorname{Vect}(\mathbb{B})$ | the four-vector reading; the eigenvalue decomposition | time and space; the spectral axis |
The quaternion construction uses the first split. The physical sectors are the second. Conflating them would identify the real part with the informational sector and the imaginary part with the material sector, and that is wrong: the material sector $\mathbb{M}_-$ has an imaginary scalar and a real vector, so it mixes the real and imaginary coefficient splits. A table of the sector action of the central $i$ is in Conventions in the Biquaternion Universe, §The Central Map and Its Restrictions.
The three splits and the three pairings are different structures with the same number. A split asks what the algebra is made of; a pairing asks how the product is grouped. The word's four factors are grouped by the pairings and typed by the splits, and neither classification implies the other.
The Cartan exponents are the one place where a split and a pairing meet. The logarithms lie in the two sectors, $\ln\tilde H=\ln r\,e_t+i\tfrac{\psi_w}{2}\vec u_{\mathrm b}\in\mathbb{M}_+$ and $\ln\tilde U=i\alpha\,e_t+\tfrac{\theta}{2}\vec u_{\mathrm r}\in\mathbb{M}_-$, but the factors do not split one per sector: $\tilde H$ is Hermitian positive, while $\tilde U$ is unitary and not an element of the linear subspace $\mathbb{M}_-$. Only the logarithms split cleanly. The direct sum is a statement about the Lie algebra of generators, not about the product. The Cartan pairing is $\tilde H\tilde U$, not $\mathbb{M}_+\times\mathbb{M}_-$.
The Action on the Sectors and the Subspaces
The four factors act by four different operations, and the table records which subspace each one preserves.
| Factor | Operation | Stable subspaces |
|---|---|---|
| scale $r$ | dilation by $r^2$ | all six |
| phase $\exp(i\alpha)$ | multiplication by the scalar | centre, vector space |
| boost $\tilde U_{\mathrm b}$ | congruence | centre, $\mathbb{M}_+$, $\mathbb{M}_-$ |
| rotor $\tilde u_{\mathrm r}$ | conjugation | all six |
The six subspaces are the centre $\mathbb{C}_{\mathbb{B}}$, the vector space $\operatorname{Vect}(\mathbb{B})$, the real quaternion space $\mathbb{H}_{\mathbb{B}}$, the imaginary quaternion space $i\mathbb{H}_{\mathbb{B}}$, and the two sectors $\mathbb{M}_\pm$. Apart from the centre, no subspace is preserved by all four factors, and that is why no single subspace characterises the word.
On the material sector the boost reads
$$ a_2=a\cosh\psi_w-b_\parallel\sinh\psi_w, \qquad b_{\parallel,2}=b_\parallel\cosh\psi_w-a\sinh\psi_w, $$
with $a$ the scalar coordinate and $b_\parallel$ the component of the vector part along the boost axis; on the informational sector the two cross terms carry the opposite sign. The difference is the sector sign. The same congruence reads as the Lorentz action on matter and as a positive congruence on the state space.
The phase at $\alpha=\pi/2$ is multiplication by $i$: it exchanges the two sectors and reverses the norm. The rotor at $\theta=2\pi$ is $-e_t$, the sign a spinor picks up after a full turn.
The Factor of Two
The word's third factor has one slot and two jobs.
| element | third factor | exponent angle | relation to the kinematic rapidity |
|---|---|---|---|
| transformation | $\tilde U_{\mathrm b}(\psi_w)=\exp(i\tfrac{\psi_w}{2}\vec u_{\mathrm b})$ | $\psi_w/2$ | $\psi_w=\psi$ |
| four-vector | $\mathcal B=\exp(-i\psi\vec u_{\mathrm b})=\tilde U_{\mathrm b}(\psi)^{-2}$ | $-\psi$ | $\psi_w=2\psi$ |
For a transformation the factor is the frame rotor, and its congruence is the boost of rapidity $\psi_w$. For a four-vector element the factor is the inverse square of the same rotor, and the element is $\tilde Q=r e^{i\alpha}\mathcal B$. The four-momentum is the case $r=mc$, $\alpha=\pi/2$, where $\tilde P=i mc\,\mathcal B$.
The doubling is between the word parameter and the kinematic rapidity of a four-vector. With $\beta=v/c$ and $\beta=\tanh\psi$,
$$ \cosh\psi_w=\frac{1+\beta^{2}}{1-\beta^{2}}=\cosh(2\psi), \qquad \gamma=\cosh\psi=\cosh\frac{\psi_w}{2}. $$
Three errors follow, and each is a check.
- Reading the word parameter $\psi_w$ of a four-vector element as its kinematic rapidity doubles the rapidity.
- Using the frame rotor $\tilde U_{\mathrm b}(\psi)$ where the four-vector factor $\mathcal B=\tilde U_{\mathrm b}(\psi)^{-2}$ belongs halves the rapidity of the four-vector: $i mc\,\tilde U_{\mathrm b}(\psi)$ is a four-vector of rapidity $\psi/2$.
- Identifying $\gamma$ with $\cosh\psi_w$ instead of $\cosh(\psi_w/2)$ overstates the energy.
The four objects called a boost are separated in Conventions in the Biquaternion Universe, §The Four Objects Called a Boost.
A Four-Vector Is Bilinear, a Spinor Is Linear
The unit factor acts on a four-vector and on a spinor by two different laws: a four-vector transforms by the congruence $X\mapsto\tilde U X\tilde U^{*}$, which is bilinear in the spinor when $X=\psi\psi^{*}$, and a spinor transforms by the left multiplication $\psi\mapsto\tilde U\psi$. The first law is quadratic in the spinor, the second linear. The sandwich is the four-vector column of the representation theory and the left multiplication is the spinor column. The polar action on the two sectors is the four-vector action; the action on the spinor module is the left multiplication, and it is the one that carries the double cover and the sign of the spinor.
A rotation through $2\pi$ gives $-\tilde U$ on the spinor and $+\mathrm{id}$ on the four-vector, because the congruence by $-\tilde U$ is the congruence by $\tilde U$. The sign is visible only in the linear action.
Physical Readings
The word is the physical description of an element, and each factor has a reading.
- The scale is the invariant length: the modulus of the four-position, $c$ for the four-velocity, $mc$ for the on-shell four-momentum. It is the only factor that survives the sandwich as a scalar, and it survives squared.
- The phase is the timelike/spacelike label and the gauge phase. It is $\pi/2$ on the timelike cone and $0$ on the spacelike sheet; it is the central $\mathrm{U}(1)$ that the sandwich does not see.
- The boost is the velocity: the axis and the word's rapidity $\psi_w$, which is twice the kinematic rapidity $\psi$ when the element is a four-vector.
- The rotor is the spin orientation: the rotation angle and axis of the spatial frame.
The pairings read the same data under different headings, and the readings are dictionaries, not derivations. The word is a coordinate statement, not a dynamics: it does not say that a system moves through the factors, that they are conserved, or that the decomposition is stable under evolution.
Limits
- The decomposition is defined off the null cone and nowhere on it. The failure is the absence of a rest frame, not a singularity of the algebra. The generalized Cartan pairing extends the amplitude–phase split to the cone with a partial isometry.
- The branch of the square root is the only convention. The interval $(-\pi/2,\pi/2]$ for the phase fixes the sign of the modulus; every other step is forced.
- The Hamilton pairing has a smaller domain than the word; the quaternion construction needs a non-vanishing real quaternion part; the Cartan pairing and the word have the widest domain. A statement proved in one pairing need not hold in another without checking the domain.
- The three splits are not the three pairings. A split decomposes the algebra; a pairing brackets a product. The real/imaginary split of the quaternion pairing is not the sector split.
Summary
- Every biquaternion off the null cone is $r\exp(i\alpha)\tilde U_{\mathrm b}\tilde u_{\mathrm r}$, with $1+1+3+3=8$ real parameters.
- The scale is the modulus of the norm, the phase is half its argument, the boost is the Hermitian positive factor and the rotor is the real quaternion factor.
- The word exists exactly off the zero-divisor cone; on the cone a lightlike four-vector has no rest frame and no boost split.
- The quaternion pairing is $\tilde Q=\tilde q\exp(i\tilde\Psi)$; the Hamilton pairing is $\tilde Q=\Rho(\cos\Theta+\vec\Xi\sin\Theta)$; the Cartan pairing is $\tilde Q=\tilde H\tilde U$.
- The three pairings are the three bracketings of the four-factor word; none adds information to it.
- The three first factors have three types — a real quaternion, a central complex scalar, a Hermitian positive element — and only the Hamilton factor is a scalar.
- The domains are not nested: $i\vec e_x+i\vec e_y$ has no quaternion construction, $e_t+\vec e_x+i\vec e_y$ has no Hamilton pairing, and $e_t+i\vec e_x$ needs the generalized Cartan form.
- The algebra has three splits — real/imaginary, Hermitian/anti-Hermitian, scalar/vector — and they are not the three pairings.
- The Cartan factors are not the two sectors; only their logarithms lie in $\mathbb{M}_\pm$.
- The scale and the rotor preserve all six remarkable subspaces, the phase preserves the centre and the vector space, and the boost preserves the centre and the two sectors.
- A four-vector transforms by the congruence and a spinor by the left multiplication; the first is bilinear in the spinor, the second linear.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $r$ | the positive scale, $r=\lvert N\rvert^{1/2}$ |
| $\alpha$ | the central phase, $\alpha=\tfrac12\arg N\in(-\pi/2,\pi/2]$ |
| $\tilde U_{\mathrm b}=\exp(i\tfrac{\psi_w}{2}\vec u_{\mathrm b})$ | the frame rotor, the word's third factor for a transformation |
| $\mathcal B=\tilde U_{\mathrm b}^{-2}=e^{-i\psi_w\vec u_{\mathrm b}}$ | the four-vector factor, the word's third factor for a four-vector element |
| $\tilde u_{\mathrm r}=\exp(\tfrac{\theta}{2}\vec u_{\mathrm r})$ | the unit real quaternion rotor |
| $\psi_w$, $\theta$ | the word parameter and the rotation angle |
| $\psi$ | the kinematic rapidity; $\psi_w=\psi$ for a transformation and $\psi_w=2\psi$ for a four-vector element |
| $\tilde q=r\tilde u_{\mathrm r}$ | the real quaternion modulus of the quaternion pairing |
| $\tilde\Psi=\alpha+\tfrac{\psi_w}{2}\vec u_{\mathrm b,2}$ | the real quaternion angle of the quaternion pairing |
| $\Rho=\sqrt N$, $\vec\Xi$, $\Theta$ | the modulus, axis and complex angle of the Hamilton pairing |
| $B=\sqrt{\langle\vec Q,\vec Q\rangle_{\natural}}$ | the vector-part root, not the four-vector factor $\mathcal B$ |
| $\tilde H=r\tilde U_{\mathrm b}$, $\tilde U=e^{i\alpha}\tilde u_{\mathrm r}$ | the Hermitian and unitary factors of the Cartan pairing |
| $\mathbb{M}_\pm$ | the Hermitian and anti-Hermitian sectors |
Further Reading
- The Polar Decomposition of Biquaternions, for the four factors, the theorem and the uniqueness.
- Exponential Polar Decomposition of Biquaternions, for the exponential word, the logarithm, the kernel and the boundary limits.
- The Quaternion Pairing of the Polar Decomposition of Biquaternions, for the real quaternion modulus and the angle.
- The Hamilton Root Pairing of the Polar Decomposition of Biquaternions, for the complex axis-angle.
- The Cartan Hermitian Pairing of the Polar Decomposition of Biquaternions, for the amplitude–phase split.
- The Polar Decomposition and Remarkable Subspaces, for the sector action, the cross-term signs and the preservation table.
- The Three Polar Factors of a Biquaternion: Frame, Mass and State, for the type-by-type comparison of the three first factors.
- The Cartan Pairing, the Amplitude and the Operator-to-State Map, for the state defined by an operator and the modular reading.
- The Hamilton Pairing: Mass as Modulus and the Complex Angle, for the mass shell and the complex angle.
- The Polar Decomposition of a Jones Matrix: Diattenuation, Retardance and Polarizers, for the Cartan pair as diattenuator and retarder, and the polarizer as its cone case.
- Conventions in the Biquaternion Universe, §The Polar Word and the Three Pairings, for the frozen notation and the domains.