The Hermitian Sylvester Equation

Introduction

This is the physics companion of The Hermitian Sylvester Equation (articles_maths/the-hermitian-sylvester-equation.md). The operator is the same, $S_{\tilde A,\tilde B}=L_{\tilde A}+R_{\tilde B}$, with the equation $\tilde AX+X\tilde B=c$; the reading is the physics of a relaxation. The three statements that carry the article are:

the anticommutator $\tilde AX+X\tilde A$ is the self-adjoint operator of a Hermitian parameter, hence the framework's energy/observable equation;
the commutator $\tilde AX-X\tilde A=[\tilde A,X]$ is the skew-adjoint operator of an anti-Hermitian parameter, hence the framework's rotation generator, and it is the derivation that carries the internal weights;
the Lyapunov operator $L_{\tilde A}+R_{\tilde A^{*}}$ is self-adjoint exactly when $\tilde A$ is Hermitian — for an anti-Hermitian $\tilde A$ it is the derivation and is skew-adjoint. This is the trap of the article, and in the physics it is the difference between a Hamiltonian evolution and a rotation.

The physics places the article next to the framework's damping, covariance and Bloch-relaxation material, and it is the operator-level form of Biquaternion Automorphisms and Derivations.

The Equation of a Relaxation

The reading of the equation. The equation

$$ \tilde AX + X\tilde B = c $$

with Hermitian $\tilde A,\tilde B$ is the fixed-point equation of a relaxation, and the operator $L_{\tilde A}+R_{\tilde B}$ is the generator of the approach to that fixed point. The mathematics article's solvability criterion — no eigenvalue of $\tilde A$ equal to the negative of an eigenvalue of $\tilde B$ — is the physical statement that the relaxation has a unique stationary point exactly when the two rates are not in resonance. The failure of the criterion is a zero mode: a non-trivial element $X$ with $\tilde AX+X\tilde B=0$ is a conserved quantity of the relaxation, and the framework's conserved modes are exactly the elements of the kernel.

The physical reading of the solution. When the relaxation is stable, the solution is the time-ordered integral of the mathematics article,

$$ X=\int_{0}^{\infty}e^{-t\tilde A}\,c\,e^{-t\tilde B}\,dt , $$

which is the damping integral of the framework: the operator decays along both channels with the rates $\tilde A$ and $\tilde B$ and the source $c$ is integrated with the two damping factors. The integral representation is the reason the corpus's damping expressions contain products of two exponentially decaying factors and not one, and it is the operator-level identity behind the corpus's linear-response and covariance formulas.

The physical reading of the spectrum. $\mathrm{spec}(L_{\tilde A}+R_{\tilde B})=\{\lambda_{i}+\mu_{j}\}$: the frequencies of the pair are the sums of the two free frequencies. The determinant formula $\prod_{i,j}(\lambda_{i}+\mu_{j})$ is the product of all pair frequencies, and its vanishing is the resonance condition. In the framework these are the two-mode energies and their sums, and the pair structure is the operator version of the addition of the two momenta of a coupled pair.

The Anticommutator: The Observable Equation

Proposition (the anticommutator is the observable). For Hermitian $\tilde A$ the operator $L_{\tilde A}+R_{\tilde A}$, $X\mapsto\{\tilde A,X\}=\tilde AX+X\tilde A$, is self-adjoint with the real spectrum $\{2\lambda_{1},\ \lambda_{1}+\lambda_{2},\ \lambda_{1}+\lambda_{2},\ 2\lambda_{2}\}$. If $\tilde A$ is definite the anticommutator operator is definite with the sign of $\tilde A$; if $\tilde A$ is indefinite it has eigenvalues of both signs and is singular exactly when $\tilde A$ has opposite eigenvalues (trace zero).

Physical reading. The anticommutator of a Hermitian observable with a Hermitian $X$ is the framework's symmetrised product, and the doubled eigenvalues $2\lambda_{i}$ are the double of the observable's values while the sum $\lambda_{1}+\lambda_{2}$ is the trace. The singularity at the indefinite element with opposite eigenvalues is the physical statement that an indefinite observable with balanced signs has a two-dimensional set of "anticommuting" elements, i.e. the framework's counterpart of the vanishing anticommutator of an operator with itself at the symmetric point.

The Hermitian Sylvester equation. For $\tilde A=\tilde B$ Hermitian the equation $\tilde AX+X\tilde A=c$ is the framework's Hermitian Sylvester equation proper, and the mathematics article's criterion becomes: uniquely solvable exactly when $\tilde A$ has no pair of opposite eigenvalues. In the physics this is the equation of the symmetrised covariance of a state under the flow generated by $\tilde A$, and its failure is the vanishing of the anticommutator for a balanced observable.

The Commutator: The Rotation Generator

Proposition (the derivation is the rotation generator). For the anti-Hermitian $\tilde A$ the inner derivation $\mathrm{ad}_{\tilde A}=L_{\tilde A}-R_{\tilde A}$ is skew-adjoint with the purely imaginary spectrum $\{\lambda_{i}-\lambda_{j}\}$, and $\tilde A\mapsto\mathrm{ad}_{\tilde A}$ is a Lie algebra homomorphism: $[\mathrm{ad}_{\tilde A},\mathrm{ad}_{\tilde B}]=\mathrm{ad}_{[\tilde A,\tilde B]}$. For Hermitian $\tilde A$ the derivation is self-adjoint instead.

Physical reading. The derivation is the infinitesimal generator of the internal rotation group of the framework, and its spectrum is the list of weights — the selection rules of the internal symmetry. The half-angle doubling appears as the difference of the eigenvalues: the internal rotation of the framework is generated by the differences of the internal charges, and the factor two of the spinor representation is the factor two of $\lambda_{i}-\lambda_{j}$ for the doubled spectrum.

The two cases in one sentence. The framework's evolutions are the anticommutator operators of the Hermitian generators (self-adjoint, real spectrum, exponentials that are not periodic) and its symmetries are the commutator operators of the anti-Hermitian generators (skew-adjoint, imaginary spectrum, exponentials that are periodic); the two are exchanged by multiplication by $i$. The framework never mixes them, and the mathematics article's trap is exactly the statement that the operator $L_{\tilde A}+R_{\tilde A^{*}}$ is not a mixture but a symmetry generator when $\tilde A$ is anti-Hermitian.

Worked Examples

A stable relaxation. $\tilde A=\tilde B=e_{0}$: $2X=c$, $X=c/2$, the identity relaxation with the two rates both equal to one. The spectrum $\{2,2,2,2\}$ is the pair spectrum of the single frequency.

A resonantly singular relaxation. $\tilde A=\tilde B=ie_{3}$ (Hermitian, eigenvalues $\{1,-1\}$): the operator $L_{ie_{3}}+R_{ie_{3}}$ has the spectrum $\{2,0,0,-2\}$ and is singular; the kernel is spanned by the two off-diagonal matrix units, which in the physics are the two coherence directions that do not relax under the balanced observable $ie_{3}$. This is the framework's resonantly conserved coherence and the exact algebraic form of a "dark state" of the corresponding channel.

A rotation generator. $\tilde A=e_{3}$ (anti-Hermitian): $L_{e_{3}}+R_{e_{3}^{\dagger}}=L_{e_{3}}-R_{e_{3}}=\mathrm{ad}_{e_{3}}$ is the derivation, skew-adjoint, spectrum $\{0,2i,-2i,0\}$: the framework's internal rotation about the third direction, of period $\pi$, with the two weights $\pm2i$ on the off-diagonal (the coherences) and the two zeros on the diagonal (the populations). The polarisation axis is the fixed direction of the rotation, and the zero weights are the invariance of the populations under it.

The identity derivation. $\mathrm{ad}_{e_{0}}=0$: the identity commutes with everything and generates no rotation. Consistently the central elements have no weights.

Summary

The Sylvester equation $\tilde AX+X\tilde B=c$ is the framework's relaxation equation: solvable uniquely exactly when the two rates are not in resonance, with the damping-integral solution $\int_{0}^{\infty}e^{-t\tilde A}ce^{-t\tilde B}dt$, and its spectrum is the pair frequencies $\lambda_{i}+\mu_{j}$. The anticommutator of a Hermitian parameter is the self-adjoint observable equation, with the doubled eigenvalues and the trace; the commutator of an anti-Hermitian parameter is the skew-adjoint rotation generator, with the weights $\lambda_{i}-\lambda_{j}$ and the Lie algebra homomorphism $\tilde A\mapsto\mathrm{ad}_{\tilde A}$. The Lyapunov operator $L_{\tilde A}+R_{\tilde A^{*}}$ is self-adjoint only for Hermitian $\tilde A$ and is the derivation for anti-Hermitian $\tilde A$; the framework's damping equations use the first and its internal rotations the second. All operator statements are proved and verified in the mathematics companion.

Summary of Notation

Symbol	Physical reading
$S_{\tilde A,\tilde B}=L_{\tilde A}+R_{\tilde B}$	Relaxation generator; $X\mapsto \tilde AX+X\tilde B$
$\{\lambda_{i}+\mu_{j}\}$	Pair frequencies; resonance at a vanishing sum
$\int_{0}^{\infty}e^{-t\tilde A}ce^{-t\tilde B}dt$	The damping integral; the stable solution
$\{\tilde A,X\}=\tilde AX+X\tilde A$	Observable equation; self-adjoint for Hermitian $\tilde A$
$[\tilde A,X]=\tilde AX-X\tilde A$	Rotation generator; skew-adjoint for anti-Hermitian $\tilde A$
$\lambda_{i}-\lambda_{j}$	The internal weights; the selection rules
$\exp(\mathrm{ad}_{e_{k}})$	Internal rotation; period $\pi$; half-angle
$\tilde A^{*}=-\tilde A$	Anti-Hermitian generator; the symmetry case