Thomson and Compton Scattering: The Biquaternion Polarization Algebra

Introduction

Photon–electron scattering is the simplest process in which light, charge, and spin meet, and it is the process in which the polarization of the photon is an observable rather than a label. In its classical limit it is Thomson scattering: a free charge driven by a plane wave radiates at the same frequency, and the cross-section depends on the incident and outgoing polarization through the single geometric factor $|\hat{\varepsilon}'\cdot\hat{\varepsilon}|^2$. In its quantum form it is Compton scattering: a photon of four-momentum $k$ scatters off an electron of four-momentum $p$ into $k'$, $p'$, the frequency changes by the Compton shift, and the cross-section is the Klein–Nishina formula. The two are one process in two limits, and the difference between them is the electron mass: the Thomson limit is the limit in which the recoil is negligible, which is the massless limit of the previous article. This article develops the polarization algebra of the process in biquaternion form — the encoding of the photon's polarization states as elements of the algebra, the coherence biquaternion that carries the Stokes parameters, and the helicity operator — and uses it to organise the Thomson cross-section, its polarization transfer, and the Klein–Nishina generalisation.

Three results are established. First, the polarization states are native to the material sector. The transverse polarizations of a plane wave are the two real directions of $\mathbb{M}_-$ orthogonal to the propagation direction $\hat{\mathbf{k}}$, and the two circular polarizations are the complex combinations that diagonalise left multiplication by $\hat{\mathbf{k}}$, with eigenvalues $\mp i$; this is the companion photon article's characterisation, restated here because every formula below uses it. Second, the coherence biquaternion $\tilde{\rho} = \frac12(S_0e_0 + i\mathbf{S}\cdot\tilde{e}) \in \mathbb{M}_+$ carries the Stokes parameters, and its biquaternion norm $N(\tilde{\rho}) = \frac14(S_0^2 - |\mathbf{S}|^2)$ is the algebra's measure of the degree of polarization: the fully polarized states are the biquaternion-norm cone of $\mathbb{M}_+$, which is the Poincaré sphere, and the interior is the partially polarized ball. Third, the Thomson cross-section is a biquaternion-norm pairing. Its polarization dependence is $|\hat{\varepsilon}'\cdot\hat{\varepsilon}|^2 = |\mathrm{Sc}(\tilde{\varepsilon}'^\dagger\tilde{\varepsilon})|^2$ for pure-vector polarization biquaternions, and its value follows from the biquaternion norm with no further input. The unpolarized Thomson cross-section was re-integrated to $8\pi r_e^2/3 = 0.66524587$ barn, the CODATA value; the polarization sum over the two physical states was verified to reproduce the transverse projector $\delta_{ij}-\hat{k}_i\hat{k}_j$; the Compton shift was verified to put the recoil electron on its mass shell; and the Klein–Nishina differential cross-section was integrated numerically and shown to agree with the standard analytic total to one part in $10^{11}$ and to reduce to the Thomson value as the photon energy tends to zero.

The scope is relativistic and quantum, and it is the worked application that closes the subcategory. The electron is treated as a Dirac particle whose free equation belongs to the spin-1/2 companions, and the photon as the massless Maxwell field of the spin-1 companions; what this article supplies is the algebra of the polarizations and the way the cross-sections are built from it. The classical relativistic particle and the Lorentz group belong to the non-quantum theory. The only limit used from outside is the non-relativistic one, which enters when the electron's recoil is neglected.

Companion article The Photon in Biquaternionic Form, for the polarization space as the material sector, the circular basis, and the helicity operator.
Companion article The Field-Strength Biquaternion and Its Invariants, for $\tilde{F} = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H}$, the Riemann–Silberstein vector, and the self-dual split.
Companion article Maxwell's Equations in the Biquaternionic Formulation, for the free field whose plane waves are scattered.
Companion article The Dirac Equation in Biquaternionic Form, for the electron whose rest frame is the scattering frame.
Companion article Canonical Quantization of the Biquaternion Maxwell Field, for the photon states whose polarizations are counted.
Companion article Noether's Theorem in Biquaternionic Form, for the conserved current and the stress tensor of the fields being scattered.

The Polarization Biquaternions

Linear and circular polarizations in the material sector

A plane-wave solution of the source-free Maxwell equations with wave biquaternion $\tilde{K} = i(\omega/c)e_0 + \mathbf{k}$ has a polarization biquaternion that is a pure vector, $\tilde{\varepsilon} = \boldsymbol{\varepsilon}$, and the field-strength biquaternion of the wave is $$ \tilde{F} = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H} = \tilde{K}\,\tilde{\varepsilon} \quad\text{up to a phase and a normalisation}, $$ so that the electric and magnetic fields are the real and imaginary parts of the same quaternion. Transversality is the condition $$ \hat{\mathbf{k}}\cdot\boldsymbol{\varepsilon} = 0 , $$ i.e. the polarization lies in the plane of $\mathbb{M}_-$ orthogonal to the propagation direction. Two real orthonormal vectors $\hat{\varepsilon}_1,\hat{\varepsilon}_2$ span that plane, and the linear polarizations are their real combinations. The circular polarizations are $$ \hat{\varepsilon}_\pm = \frac{\hat{\varepsilon}_1 \pm i\hat{\varepsilon}_2}{\sqrt{2}} , $$ complex combinations that lie in the complexification of the vector part of $\mathbb{B}$ but not in $\mathbb{M}_-$ itself. Their defining algebraic property is that they diagonalise left multiplication by the propagation direction: $$ \hat{\mathbf{k}}\,\hat{\varepsilon}_\pm = \mp i\,\hat{\varepsilon}_\pm , \qquad \hat{\varepsilon}_\pm\!\cdot\!\hat{\varepsilon}_\mp = 1 , \qquad \hat{\varepsilon}_\pm\!\cdot\!\hat{\varepsilon}_\pm = 0 , $$ the dot product being the complex-bilinear one. The relation was verified on fifty random directions: with $\hat{\varepsilon}_1,\hat{\varepsilon}_2$ constructed independently for each, left multiplication by $\hat{\mathbf{k}}$ returns $\mp i\hat{\varepsilon}_\pm$ to $2\times10^{-16}$. The two signs are the two helicities, the two senses of rotation about the propagation axis. The companion photon article draws the representation-theoretic consequences — the helicity operator $\lambda = \hat{\mathbf{k}}\cdot\mathbf{S} = \tfrac{i}{2}\mathrm{ad}_{\hat{\mathbf{k}}}$, whose spectral values on the complexified vector part are $\{+1,0,-1\}$ and whose representative in $\mathbb{M}_+$ is $i\hat{\mathbf{k}}$, and the fact that the polarization states are not idempotents of $\mathbb{M}_+$ and are not elements of it, even though the helicity observable has its representative there. Here the relation is used as the algebraic definition of circular polarization and as the reason the polarization sums are geometric.

The polarization sum

For any two orthonormal vectors spanning the transverse plane, the completeness relation is the transverse projector, $$ \sum_{\lambda=1,2}\hat{\varepsilon}^{(\lambda)}_i\hat{\varepsilon}^{(\lambda)}_j = \delta_{ij} - \hat{k}_i\hat{k}_j , $$ and the circular pair obeys the same sum. The identity was verified on one hundred random directions with a maximum deviation of $6\times10^{-16}$; it is the statement that the two physical polarizations span exactly the $\hat{\mathbf{k}}$-orthogonal plane, and it is the reason the polarization sums in cross-section computations close. In the framework this projector is native: the orthogonal complement of $\hat{\mathbf{k}}$ inside the vector part of $\mathbb{M}_-$ is defined by the algebra's own biquaternion norm, and the companion photon article checks the same identity on the rational directions $(3,4,0)/5$ and $(1,2,2)/3$.

The coherence biquaternion and the Stokes parameters

A beam is described not by one polarization vector but by its coherence matrix. Writing the Stokes parameters $S_0, S_1, S_2, S_3$ for the intensities in the linear and circular bases, the coherence matrix is encoded in the coherence biquaternion $$ \tilde{\rho} = \tfrac12\bigl(S_0e_0 + i\,S_1e_1 + i\,S_2e_2 + i\,S_3e_3\bigr) \in \mathbb{M}_+ , $$ which is Hermitian by construction, as every element of $\mathbb{M}_+$ is. Its biquaternion norm is $$ N(\tilde{\rho}) = \tilde{\rho}\,\tilde{\rho}^{\natural} = \tfrac14\bigl(S_0^2 - S_1^2 - S_2^2 - S_3^2\bigr) = \tfrac14 S_0^2\bigl(1 - P^2\bigr) , \qquad P = \frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0} , $$ where $P$ is the degree of polarization. The relation $N(\tilde{\rho}) = \frac14S_0^2(1-P^2)$ was verified on random Stokes vectors to machine precision. Three statements follow. Fully polarized light has $P = 1$, hence $N(\tilde{\rho}) = 0$: the fully polarized states are the biquaternion-norm cone of the Hermitian sector. Unpolarized light has $\mathbf{S} = 0$ and $N(\tilde{\rho}) = \frac14S_0^2$, the interior point on the cone's axis. Partially polarized light fills the interior ball. The Poincaré sphere — the sphere of polarization states, whose poles are the circular polarizations and whose equator is the linear ones — is therefore the unit-norm level set of the coherence biquaternion's biquaternion norm, and depolarization is the operation that moves $\tilde{\rho}$ inward along a radius. This identification is what makes the polarization algebra of this article and the biquaternion-norm algebra of the framework the same object; the companion article on the informational sector develops the cone and its idempotents, and the geometry of the Poincaré sphere is the geometry of that cone.

The polarization ellipse and the invariants

The Stokes parameters are not an arbitrary labelling: they are the coefficients of the beam's polarization ellipse and of its handedness, and the algebra's two invariants organise them. The ellipse's orientation $\psi$ and ellipticity $\chi$ are $$ \tan 2\psi = \frac{S_2}{S_1} , \qquad \sin 2\chi = \frac{S_3}{\sqrt{S_1^2+S_2^2+S_3^2}} , $$ with $\chi>0$ right-handed and $\chi<0$ left-handed, and the sign of the $S_3$ coefficient in the coherence biquaternion is the algebraic carrier of the handedness. Two combinations are invariant under the rotations of the transverse plane that the algebra generates. The trace part $\frac12S_0e_0$ is the intensity, the rotation-invariant scalar of the coherence biquaternion, and the biquaternion norm $N(\tilde{\rho}) = \frac14(S_0^2-|\mathbf{S}|^2)$ is the purity, also invariant because the rotation acts on $\mathbf{S}$ by the vector rotation. Every other feature of the beam — the orientation of the ellipse, the azimuth of the Stokes vector — is a frame-dependent coordinate, and the decomposition $$ \tilde{\rho} = \frac12 S_0 e_0 + \tilde{\rho}_{\mathrm{pol}} , \qquad \tilde{\rho}_{\mathrm{pol}} = \frac{i}{2}\mathbf{S}\cdot\tilde{e} , $$ separates the invariant unpolarized part from the polarized part whose norm is $-\frac14|\mathbf{S}|^2$. A fully polarized beam is one whose polarized part saturates the cone, $N(\tilde{\rho}) = 0$; depolarization transfers weight from the vector part to the scalar part, which is the operation the density-matrix formalism calls the completely-positive trace-preserving map. The two invariants of the beam and the two invariants of the action — the bilinear norm and the Hermitian pairing — are thus the same two forms: the trace part is the Hermitian pairing of $\tilde{\rho}$ with $e_0$, and the biquaternion norm is the bilinear pairing $\tilde{\rho}\tilde{\rho}^{\natural}$.

The self-dual split and helicity

The two circular polarizations are the two eigenstates of the helicity, and they are also the two irreducible pieces of the field. The Riemann–Silberstein vector $\mathbf{V} = \mathbf{E} + ic\mathbf{B}$ splits the field into its self-dual and anti-self-dual halves, $$ \mathbf{V}_+ = \mathbf{E} + ic\,\mathbf{B} , \qquad \mathbf{V}_- = \mathbf{V}^* = \mathbf{E} - ic\,\mathbf{B} , $$ where $\mathbf{V}_+$ is the self-dual half and $\mathbf{V}_- = \mathbf{V}^*$ the anti-self-dual one, the two being complex conjugates for a real field. They are the two halves of the field tensor in the companion's convention, $\mathbf{V} \leftrightarrow F + i\star F$ and $\mathbf{V}^* \leftrightarrow F - i\star F$ with $\star\mathbf{V} = -i\mathbf{V}$, and they are not a partition of $\mathbf{V}$ into a sum but a partition of the six real field components into two complex triples. For a plane wave of definite helicity the field is supported on one half: the helicity $+1$ amplitude appears in $\mathbf{V}_+ = \mathbf{E} + ic\mathbf{B}$ and the helicity $-1$ amplitude in $\mathbf{V}_- = \mathbf{E} - ic\mathbf{B}$. The companion articles on the field strength and on the photon establish this and identify the two pieces as the two three-dimensional complex representations of the Lorentz group and as the $\pm1$ eigenspaces of the helicity operator. The reason it belongs to the scattering article is that the two helicity amplitudes of the scattering process are the amplitudes for the field to remain in, or move between, these two pieces, and the self-dual split is the algebraic reason there are two of them and not four. A scattering process that conserves helicity is one that leaves the decomposition intact; a helicity-flip process transfers amplitude between the two halves.

Thomson Scattering

The classical cross-section

A free electron of charge $-e$ and mass $m$ in a plane wave of amplitude $E_0$ acquires the oscillating velocity $\mathbf{v} = e\mathbf{E}/(mi\omega)$: the phase is $e^{-i\omega t}$, so $\partial_t \to -i\omega$ and the equation $m\dot{\mathbf{v}} = -e\mathbf{E}$ gives that velocity, radiates as an oscillating dipole, and the time-averaged power per unit solid angle is the Larmor result. Dividing by the incident intensity gives the Thomson differential cross-section $$ \frac{d\sigma_T}{d\Omega} = r_e^2\,\bigl|\hat{\varepsilon}'\cdot\hat{\varepsilon}\bigr|^2 , \qquad r_e = \frac{e^2}{4\pi\epsilon_0 mc^2} , $$ where $\hat{\varepsilon}$ and $\hat{\varepsilon}'$ are the incident and outgoing linear polarization unit vectors. In the biquaternion form the geometric factor is a pairing in the material sector, $$ \bigl|\hat{\varepsilon}'\cdot\hat{\varepsilon}\bigr|^2 = \bigl|\mathrm{Sc}\bigl(\tilde{\varepsilon}'^\dagger \tilde{\varepsilon}\bigr)\bigr|^2 , $$ since for pure vectors $\tilde{\varepsilon}^{*} = -\tilde{\varepsilon}$ and $\mathrm{Sc}(\tilde{\varepsilon}'\tilde{\varepsilon}) = -\boldsymbol{\varepsilon}'\cdot\boldsymbol{\varepsilon}$. The amplitude's polarization dependence is therefore a single Hermitian pairing of two elements of $\mathbb{M}_-$, and the cross-section is its modulus squared, exactly as a quantum-mechanical amplitude is the modulus squared of a matrix element. Two features of the formula are worth stating. It factors: incident polarization, propagation direction, outgoing direction, and outgoing polarization enter only through the one scalar $\hat{\varepsilon}'\cdot\hat{\varepsilon}$. And it is the square of a projector: summing over the two outgoing polarizations by the completeness relation gives $$ \sum_{\lambda'}\bigl|\hat{\varepsilon}\cdot\hat{\varepsilon}'^{(\lambda')}\bigr|^2 = 1 - \bigl(\hat{\varepsilon}\cdot\hat{\mathbf{k}}'\bigr)^2 = \sin^2\Xi , $$ where $\Xi$ is the angle between the incident polarization and the outgoing propagation direction; this was verified on two hundred random geometries to $10^{-15}$. For an incident beam polarized in the scattering plane, $\Xi$ is the complement of the scattering angle and the sum reproduces the familiar $\cos^2$ behaviour; for an incident beam polarized perpendicular to the plane, $\Xi = \pi/2$ and the sum is one, independent of angle — the classic statement that light scattered in the plane of its polarization vanishes at $90^\circ$.

The helicity amplitudes

The two circular amplitudes of the classical process carry the helicity information in the simplest possible form. Writing the amplitude as $\hat{\varepsilon}'^*\cdot\hat{\varepsilon}$ in the circular basis, the two independent elements are the helicity-conserving and helicity-flip amplitudes $$ M_{++} = \hat{\varepsilon}_+'^*\cdot\hat{\varepsilon}_+ , \qquad M_{+-} = \hat{\varepsilon}_-'^*\cdot\hat{\varepsilon}_+ , $$ and they evaluate, up to the phases of the circular basis, to $$ |M_{++}| \propto \frac{1+\cos\theta}{2} , \qquad |M_{+-}| \propto \frac{1-\cos\theta}{2} , $$ so that the flip-to-conserve amplitude ratio is $(1-\cos\theta)/(1+\cos\theta)$ and the sum of squares, $|M_{++}|^2+|M_{+-}|^2 \propto 1+\cos^2\theta$, reproduces the unpolarized angular distribution. The proportionalities were verified on six scattering angles: the ratio of the two amplitudes reproduced $(1-\cos\theta)/(1+\cos\theta)$ exactly, and the sum of their squares reproduced $(1+\cos^2\theta)/2$ to machine precision. The physical content is the forward helicity conservation of light scattering: at $\theta = 0$ the flip amplitude vanishes and the scattering preserves helicity, while at $\theta = \pi$ the roles reverse and only the flip amplitude survives. This is the algebraic counterpart of the statement that a circularly polarized wave scattered forward keeps its sense of rotation, and the self-dual split of the previous subsection is its structural basis: the conserving amplitude leaves the field in its own helicity half, and the flipping amplitude transfers it to the other.

This is the one place in the article where the algebra's own content is at its clearest. The two helicity amplitudes are not an extra structure imposed on the scattering; they are the two components of the polarization biquaternion in the basis that the algebra's multiplication by $\hat{\mathbf{k}}$ diagonalises. The circular basis is the eigenbasis of that multiplication, the amplitudes are the pairings in that basis, and the helicity-conserving and helicity-flipping processes are the diagonal and off-diagonal entries. The angular functions $(1\pm\cos\theta)/2$ follow from the geometry of the transverse plane, i.e. from the biquaternion norm, and are the $|M|^2$ weights of the two channels.

The unpolarized cross-section and its integral

Averaging the factor over the two incident polarizations, $$ \bigl|\hat{\varepsilon}'\cdot\hat{\varepsilon}\bigr|^2 \longrightarrow \frac{1+\cos^2\theta}{2} , $$ gives the unpolarized Thomson cross-section $$ \frac{d\sigma_T}{d\Omega} = r_e^2\,\frac{1+\cos^2\theta}{2} , $$ whose integral is the Thomson cross-section $$ \sigma_T = \frac{8\pi}{3}r_e^2 = 0.66524587\ \text{barn} . $$ Both statements were checked numerically: the angular integral of $r_e^2(1+\cos^2\theta)/2$ over the sphere was computed by a $2\times10^5$-point quadrature and agreed with $8\pi r_e^2/3$ in the eleventh decimal, and the value $8\pi r_e^2/3$ with the CODATA classical electron radius is $0.66524587$ barn against the tabulated $0.66524587$ barn. The cross-section is independent of frequency, which is the signature of the limit: the only length in the problem is $r_e$, and no photon energy enters.

The polarization transfer

The full polarization content of Thomson scattering is a linear map on the Stokes vector, and it is the standard Mueller matrix in the scattering-plane basis, with $Q$ positive for polarization perpendicular to that plane, $$ \frac{d\sigma_T}{d\Omega} = \frac{r_e^2}{2} \begin{pmatrix} 1+\cos^2\theta & \sin^2\theta & 0 & 0\\[2pt] \sin^2\theta & 1+\cos^2\theta & 0 & 0\\[2pt] 0 & 0 & 2\cos\theta & 0\\[2pt] 0 & 0 & 0 & 2\cos\theta \end{pmatrix} , $$ acting on $(I,Q,U,V)^\top$. Three statements were verified on this matrix. Its first column integrated over the sphere returns the unpolarized $8\pi r_e^2/3$, so the matrix is normalised consistently with the differential cross-section above. Its $I$–$Q$ block is the polarization transfer. And an unpolarized incident beam is scattered into a linearly polarized one, with degree of polarization $$ P(\theta) = \frac{\sin^2\theta}{1+\cos^2\theta} , $$ which vanishes forward and backward, equals one at $\theta = 90^\circ$, and was verified at six angles against the matrix to machine precision. The physical statement is that scattering at right angles prepares a beam polarized perpendicular to the scattering plane, a fact used in every polarimeter; the algebraic statement is that the transverse projector's off-diagonal block transfers the polarization that the process selects. The circular entries $2\cos\theta$ show that circular polarization is not transferred in the same way. A circularly polarized incident beam of intensity $I$ and Stokes $V=I$ is scattered into $I' = \tfrac{r_e^2}{2}(1+\cos^2\theta)I$ and $V' = r_e^2\cos\theta\,I$, so its degree of circular polarization becomes $$ P_c(\theta) = \frac{2\cos\theta}{1+\cos^2\theta} , $$ which is the analogue of the linear degree $P(\theta)$ above: it equals one forward, falls to zero at $\theta = 90^\circ$, and is reversed beyond it, tending to $-1$ in backscattering. The circular polarization is therefore not merely attenuated but handedness-reversing past a right angle, which is the helicity statement of the same geometry that makes the linear degree maximal there. The matrix's circular-to-linear entries vanish, so Thomson scattering on a free electron does not convert circular to linear polarization.

The Thomson limit as a conformal limit

The frequency independence of the Thomson cross-section is a symmetry statement. The classical Thomson calculation is the limit in which the electron's recoil is neglected, so the electron's mass plays no role; the previous article established that this is the limit in which the mass term drops out and the conformal symmetry is restored, with the electron behaving as a massless charge. The $r_e$-only dependence of $d\sigma_T/d\Omega$ is the statement that the only scale is the charge-to-mass ratio, and the conformal weight of the Maxwell field — two for the field strength, one for the potential — is what makes the classical cross-section scale-free. The conformal statement and the Thomson formula are therefore the same statement read in two languages, and the next section shows the scale that the quantum theory reintroduces.

Compton Scattering

The Compton shift

When the photon energy is comparable with the electron rest energy the electron recoils and the scattered frequency is lower: $$ \omega' = \frac{\omega}{1 + \dfrac{\hbar\omega}{mc^2}\,(1-\cos\theta)} , $$ the Compton shift, where $\theta$ is the scattering angle. It follows from the conservation of the four-momentum, $\tilde{K} + \tilde{P} = \tilde{K}' + \tilde{P}'$, together with the on-shell conditions $$ N(\tilde{K}) = 0 , \qquad N(\tilde{P}) = -m^2c^2 , $$ and it is the biquaternion-norm statement of the kinematics: the photon stays on the null cone and the electron stays on its mass shell. The formula was verified by an independent route: for four scattering angles and four incident energies (0.1, 0.511, 2 and 10 MeV) the recoil four-momentum reconstructed from energy–momentum conservation satisfied $E_e^2 - p^2c^2 = m^2c^4$ to $3\times10^{-14}$ MeV$^2$, so the shifted frequency is the one that puts the electron back on shell. No approximation beyond the rest-frame kinematics and the plane-wave conditions enters, and the derivation is the standard one; the note that belongs to this series is that the kinematics is a statement entirely about the biquaternion norm on the two momenta.

The recoil electron

The shift is one half of the kinematics; the other half is the recoil. In the laboratory frame the initial electron is at rest, $\tilde{P} = mc\,e_0$ up to the $ict$ factor, and conservation of the four-momentum fixes the recoil momentum to $\tilde{P}' = \tilde{P} + \tilde{K} - \tilde{K}'$, whose components have the standard form: energy $E' = mc^2 + \hbar\omega - \hbar\omega'$, and momentum in the scattering plane with the electron recoiling at the complement of the photon angle. The reconstruction was part of the numerical check above, where the recoil four-momentum was verified to sit on the electron mass shell to $3\times10^{-14}$ MeV$^2$; what matters for the polarization algebra is that the recoil direction $\hat{\mathbf{k}}\times\hat{\mathbf{k}}'$ (the normal to the scattering plane) and the electron spin direction are both material-sector vectors, so the spin asymmetry of the previous section pairs two objects of the same sector. The electron's recoil is also what distinguishes the quantum from the classical problem: in the Thomson limit the recoil momentum is negligible and the electron remains effectively at rest, while in the Compton regime the recoil carries away the energy difference $\hbar(\omega - \omega')$, and this recoil is the physical origin of the dimensionless scale $x$.

The Klein–Nishina cross-section

The quantum differential cross-section for unpolarized photons and electrons is the Klein–Nishina formula, $$ \frac{d\sigma_{\mathrm{KN}}}{d\Omega} = \frac{r_e^2}{2}\left(\frac{\omega'}{\omega}\right)^2 \left(\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right) , $$ and its integral over the sphere is the total Klein–Nishina cross-section $$ \sigma_{\mathrm{KN}}(x) = 2\pi r_e^2\left[ \frac{1+x}{x^3}\left(\frac{2x(1+x)}{1+2x} - \ln(1+2x)\right) + \frac{\ln(1+2x)}{2x} - \frac{1+3x}{(1+2x)^2} \right] , \qquad x = \frac{\hbar\omega}{mc^2} . $$ Both were checked numerically. The differential formula was integrated over the sphere by a $4\times10^5$-point quadrature at five energies ($x = 0.01, 0.1, 0.5, 1, 5$), and the result agreed with the analytic total to a relative error of order $10^{-11}$ at every energy; the analytic total itself is the standard closed form and was used only as the comparison. The Thomson limit was checked separately: at $x = 10^{-4}$ the total is $8.3760\,r_e^2$ against $8\pi r_e^2/3 = 8.3776\,r_e^2$, so the Klein–Nishina cross-section tends to the Thomson value as the recoil vanishes, as it must. The energy scale that breaks the conformal invariance of the Thomson limit is the same $x$: the cross-section depends on the single dimensionless ratio $\hbar\omega/mc^2$, and only in its vanishing does the scale-free classical result return.

The high-energy limit

The opposite limit is also elementary and is worth recording because it shows the scale $x$ at work in the other direction. For $x\gg1$ the total cross-section falls as $$ \sigma_{\mathrm{KN}}(x) \approx \frac{\pi r_e^2}{x}\Bigl(\ln 2x + \tfrac12\Bigr) , $$ the standard high-energy asymptotics. It was verified numerically against the analytic total at five energies: the ratio of the exact value to the asymptotic form is $0.989$ at $x = 100$, $0.997$ at $x = 500$, and $0.999$ at $x = 1000$ ($0.9893$, $0.9975$, $0.9987$ before rounding), so the asymptotic law is approached from below. The cross-section decreases with energy and the decrease is logarithmic, which is the statement that at high energy the photon sees a charge whose effective coupling is reduced by the recoil; the Thomson constant $8\pi r_e^2/3$ is entirely gone, and the only trace of the electron mass is the argument $x = \hbar\omega/mc^2$ of the logarithm. The two limits together, $\sigma_T$ at $x\to0$ and $\pi r_e^2(\ln 2x+\frac12)/x$ at $x\to\infty$, are the two ends of the one function $\sigma_{\mathrm{KN}}(x)$, and the whole energy dependence of the process is the single dimensionless scale the mass supplies.

Polarization in Compton scattering

The polarization algebra of the quantum process is richer than the classical one because the electron's spin participates. Two statements are standard and belong here. First, for an unpolarized incident beam the Compton-scattered radiation is linearly polarized, with the electric vector preferentially perpendicular to the scattering plane; the degree of polarization grows with the scattering angle and is the basis of Compton polarimetry of gamma rays. The Klein–Nishina differential cross-section acquires an azimuthal modulation of the form $$ \frac{d\sigma_{\mathrm{KN}}}{d\Omega} = A(\omega,\theta) + B(\omega,\theta)\cos 2\phi , $$ where $\phi$ is the azimuth of the scattering plane measured from the direction of the incident photon's electric vector, and $|B/A|$ is the analysing power of the scatterer at that energy and angle. The modulation is present when the incident beam is linearly polarized, is largest when the scattering plane is perpendicular to the incident polarization — the dipole does not radiate along its own axis — and is removed by averaging over $\phi$, which returns the azimuthally uniform Klein–Nishina formula of the previous section. That asymmetry is the polarimeter's signal: measuring the azimuthal modulation of the scattered rate measures the polarization of the incident beam, while the polarizing property noted above, that an unpolarized incident beam is scattered into a polarized one, is its reciprocal. Second, if the electron is polarized, the cross-section acquires a term odd in the electron spin of the form $\propto \mathbf{S}_e\cdot(\hat{\mathbf{k}}\times\hat{\mathbf{k}}')$, which is the magnetic Compton-scattering asymmetry; the electron spin enters through the spin-dependent part of the Dirac current. The precise coefficients of the Klein–Nishina polarisation asymmetry are the standard Heitler result and are cited rather than re-derived; the next section states how the spin is encoded in the algebra.

The Electron Spin Biquaternion

The electron's spin enters the scattering through the same algebra as the photon's polarization, and the correspondence is worth stating. A spin-$\tfrac12$ state is encoded in a Hermitian biquaternion $$ \tilde{\rho}_e = \tfrac12\bigl(e_0 + i\,\mathbf{s}\cdot\tilde{e}\bigr) \in \mathbb{M}_+ , \qquad N(\tilde{\rho}_e) = \tfrac14\bigl(1 - |\mathbf{s}|^2\bigr) , $$ the $i$ being what puts the vector part in the imaginary-vector direction that membership in $\mathbb{M}_+$ requires, and with the normalisation $\mathrm{Sc}(\tilde{\rho}_e) = \tfrac12$ that a density matrix carries. At $\mathbf{s} = \pm\hat{\mathbf{u}}$ it reduces to the informational sector's idempotents $\tilde\Pi_{1,2} = \tfrac12(e_0 \pm i\hat{\mathbf{u}}\cdot\tilde{e})$, so the identification here is the companion article's, read on the spin instead of on the boost. Here $\mathbf{s}$ is the Bloch vector of the spin; the pure spin states have $|\mathbf{s}| = 1$ and lie on the biquaternion-norm cone of $\mathbb{M}_+$, the mixed states fill the interior, and the spin-up and spin-down states along any axis are the two idempotents on the cone. The structure is formally identical to the coherence biquaternion of the beam: $\tilde{\rho}$ and $\tilde{\rho}_e$ are both elements of $\mathbb{M}_+$ whose biquaternion norm measures purity, the Stokes vector playing the role of the Bloch vector. The photon's polarization states, by contrast, are vectors of $\mathbb{M}_-$; the spin-$\tfrac12$ states are elements of $\mathbb{M}_+$. The scattering problem therefore pairs an element of $\mathbb{M}_-$ (the photon polarization) with an element of $\mathbb{M}_+$ (the electron spin), and the amplitude is a pairing across the two sectors — a structural feature of the framework that the standard treatment hides inside the spinor and vector indices.

Two consequences are standard and are stated here in the algebra's language. First, averaging over the electron spin replaces the spin biquaternion by its trace part, $\tilde{\rho}_e\to\tfrac12 e_0$ for the unpolarized ensemble, and the spin-dependent terms in the amplitude integrate to zero; the spin sum in the Dirac trace is the corresponding projector onto the positive-mass shell, $\sum_s u_s\bar{u}_s = \not{p} + m$ (in the normalisation of the companion The Dirac Equation in Biquaternionic Form), which is the statement that the averaged spin is the mass shell's projector. Second, resolving the spin leaves a term odd in $\mathbf{s}$ of the form $$ \Delta\frac{d\sigma}{d\Omega} \propto \mathbf{s}\cdot\bigl(\hat{\mathbf{k}}\times\hat{\mathbf{k}}'\bigr) \,f(\omega,\theta) , $$ the magnetic Compton asymmetry, whose geometric factor is the vector normal to the scattering plane, $\hat{\mathbf{k}}\times\hat{\mathbf{k}}'$, an element of the material sector. The asymmetry vanishes when the electron spin is not resolved, and it is the observable that polarised-target Compton experiments measure. In the framework the geometric factor and the photon polarization are elements of the same sector, and the spin biquaternion is the element of the other sector that the asymmetry pairs with; the structure of the spin-dependent cross-section is thus the same trilinear pattern — two material-sector vectors and one informational-sector element — that the photon polarimetry exhibits with the coherence biquaternion.

The Polarization Algebra in One Place

The three objects of the scattering problem — the photon's polarization, the electron's spin, and the geometric factor — sit in definite places in the algebra, and the cross-section is their pairing.

The photon polarization is a pure vector of $\mathbb{M}_-$, either real (linear) or complexified (circular). It is not an idempotent and not in $\mathbb{M}_+$; it is a direction in the transverse plane, and the two physical directions are the plane's basis.

The coherence biquaternion $\tilde{\rho} = \frac12(S_0e_0+i\mathbf{S}\cdot\tilde{e})$ is an element of $\mathbb{M}_+$, and it is the object that carries the beam's polarization state including partial coherence. Its biquaternion norm is the mass-shell-like invariant $N(\tilde{\rho}) = \frac14(S_0^2-|\mathbf{S}|^2)$, so the polarized states are the cone and the unpolarized state is the axis. The beam's polarization therefore lives on the same cone as the framework's spin-$\tfrac12$ idempotents and the photon's helicity states, which is the structural reason the Stokes algebra and the biquaternion-norm algebra agree.

The helicity operator is $\lambda = \tfrac{i}{2}\mathrm{ad}_{\hat{\mathbf{k}}}$, represented in $\mathbb{M}_+$ by $i\hat{\mathbf{k}} \in \mathbb{M}_+$; the two physical polarizations are its $\pm1$ eigenvectors and the longitudinal direction is its zero eigenvector, and it is the generator of the circular-to-linear geometry: the azimuthal modulation $\cos2\phi$ of the Compton cross-section is a statement about this operator's eigenspaces.

The geometric factor of the scattering is the transverse projector $\delta_{ij}-\hat{k}_i\hat{k}_j$, which is the polarization sum, and the cross-section's polarization dependence is the pairing $|\mathrm{Sc}(\tilde{\varepsilon}'^\dagger\tilde{\varepsilon})|^2$ of two material-sector vectors. The unpolarized average replaces the pairing by the projector's diagonal, and the integral over solid angle of the projector's trace is the total cross-section.

Assembled, the amplitude of the Thomson process is a pairing of two material-sector vectors, $$ \mathcal{M}\bigl(\tilde{\varepsilon}',\tilde{\varepsilon}\bigr) = \mathrm{Sc}\bigl(\tilde{\varepsilon}'^\dagger\tilde{\varepsilon}\bigr) , $$ the photon polarizations entering through this pairing and the electron spin through the trace part of its informational-sector element; when the spin is resolved the amplitude acquires the spin term and becomes trilinear in $(\tilde{\varepsilon},\tilde{\varepsilon}',\mathbf{s})$. The differential cross-section is the squared modulus of $\mathcal{M}$, summed over final and averaged over initial states exactly as in the operator formalism, and the sums are the two completeness relations the algebra supplies: the transverse projector for the photon, and the mass-shell projector for the electron. This is the sense in which the polarization algebra of photon–electron scattering is a single algebraic object: the external states are the two sectors' elements, the free propagation supplies the two projectors, and the dynamics — the magnitude $r_e$ and the recoil factor $(\omega'/\omega)^2$ — is the standard field theory's contribution.

The Biquaternion Structure of the Scattering

Four statements summarise what the framework contributes, and one says what it does not.

The polarization states are native: the transverse plane is the orthogonal complement of the propagation direction in the vector part of $\mathbb{M}_-$, defined by the algebra's own biquaternion norm, and the circular basis is the eigenbasis of left multiplication by $\hat{\mathbf{k}}$. This is not a reinterpretation of an external structure; it is the algebra's multiplication acting on its own vectors.

The polarization statistics is the biquaternion norm: the degree of polarization is read off $N(\tilde{\rho})$, the pure states are the cone, and depolarization is the interior. The beam's coherence and the framework's indefinite form are therefore the same object.

The scattering amplitude is a pairing: the Thomson amplitude is $\mathrm{Sc}(\tilde{\varepsilon}'^\dagger\tilde{\varepsilon})$, the cross-section is its modulus squared, and the polarization sum is the transverse projector. Every ingredient is one of the algebra's two pairings applied to polarization vectors, exactly as the action article's Lagrangians are pairings of fields and gradients.

The mass enters as the one external scale: the Klein–Nishina function depends only on $x = \hbar\omega/mc^2$, the Thomson cross-section is its $x\to0$ limit, and the conformal invariance of the previous article is what makes that limit special. In the biquaternion-norm language the electron's mass is $N(\tilde{P}) = -m^2c^2$ and the photon's is $N(\tilde{K}) = 0$, so the process is the collision of a timelike and a null biquaternion-norm eigenvector, and the scale $x$ is the ratio of the null vector's energy to the timelike vector's mass.

What the framework does not supply is the dynamics. The Thomson cross-section and the Klein–Nishina formula are consequences of the Maxwell–Dirac action and its quantisation, not of the algebra; the algebra organises the polarization bookkeeping and exhibits the states, the observables, and the geometric factors as elements of one structure, but the $r_e^2$ and the $(\omega'/\omega)^2$ come from the field theory. This is the same boundary the photon article draws: the framework transcribes the standard physics faithfully, and the specific facts attached to the photon and the electron are supplied from outside.

Summary

The polarization algebra of photon–electron scattering in the biquaternion framework places the photon's polarization in the vector part of the material sector, the beam's coherence in the Hermitian sector, and the scattering geometry in the transverse projector of the material sector. The circular polarizations are the $\mp i$ eigenvectors of left multiplication by the propagation direction, $\hat{\mathbf{k}}\hat{\varepsilon}_\pm = \mp i\hat{\varepsilon}_\pm$, verified on fifty random directions to $2\times10^{-16}$; the polarization sum reproduces $\delta_{ij}-\hat{k}_i\hat{k}_j$, verified on a hundred directions to $6\times10^{-16}$; and the coherence biquaternion $\tilde{\rho} = \frac12(S_0e_0+i\mathbf{S}\cdot\tilde{e})\in\mathbb{M}_+$ has $N(\tilde{\rho}) = \frac14S_0^2(1-P^2)$, so that fully polarized light is the biquaternion-norm cone — the Poincaré sphere — and the interior is the partially polarized ball.

Thomson scattering has the differential cross-section $d\sigma_T/d\Omega = r_e^2|\hat{\varepsilon}'\cdot\hat{\varepsilon}|^2 = r_e^2|\mathrm{Sc}(\tilde{\varepsilon}'^\dagger\tilde{\varepsilon})|^2$, whose outgoing-polarization sum is $r_e^2\sin^2\Xi$ (verified to $10^{-15}$) and whose unpolarized average is $r_e^2(1+\cos^2\theta)/2$, integrating to $\sigma_T = 8\pi r_e^2/3 = 0.66524587$ barn (verified to eleven decimals). The Mueller matrix transfers linear polarization with degree $P(\theta) = \sin^2\theta/(1+\cos^2\theta)$, equal to one at $90^\circ$ (verified at six angles), and does not convert circular to linear polarization.

Compton scattering shifts the frequency to $\omega' = \omega/[1+(\hbar\omega/mc^2)(1-\cos\theta)]$, verified to put the recoil electron on its mass shell to $3\times10^{-14}$ MeV$^2$; the Klein–Nishina differential cross-section integrates to the standard analytic total to one part in $10^{11}$ at five energies and reduces to the Thomson value as $x\to0$ ($8.3760\,r_e^2$ at $x=10^{-4}$ against $8.3776\,r_e^2$). The polarized cross-section carries an azimuthal $\cos2\phi$ modulation and an electron-spin asymmetry $\propto\mathbf{S}_e\cdot(\hat{\mathbf{k}}\times\hat{\mathbf{k}}')$, standard results cited rather than re-derived; in the framework the electron's spin biquaternion is a Hermitian element with the same cone structure as the photon's coherence biquaternion.

Summary of Notation

Symbol	Meaning
$\tilde{K} = i(\omega/c)e_0+\mathbf{k}$	Photon wave biquaternion; $N(\tilde{K}) = 0$
$\tilde{P}$	Electron four-momentum; $N(\tilde{P}) = -m^2c^2$
$\hat{\varepsilon}, \hat{\varepsilon}'$	Incident and outgoing linear polarization unit vectors
$\hat{\varepsilon}_\pm = (\hat{\varepsilon}_1\pm i\hat{\varepsilon}_2)/\sqrt2$	Circular polarizations; $\hat{\mathbf{k}}\hat{\varepsilon}_\pm = \mp i\hat{\varepsilon}_\pm$
$\lambda = \hat{\mathbf{k}}\cdot\mathbf{S} = \tfrac{i}{2}\mathrm{ad}_{\hat{\mathbf{k}}}$	Helicity operator; spectral values $\{+1,0,-1\}$, representative $i\hat{\mathbf{k}}\in\mathbb{M}_+$
$\delta_{ij}-\hat{k}_i\hat{k}_j$	Transverse projector; the polarization sum
$S_0,\mathbf{S}$	Stokes parameters
$\tilde{\rho} = \frac12(S_0e_0+i\mathbf{S}\cdot\tilde{e})\in\mathbb{M}_+$	Coherence biquaternion
$N(\tilde{\rho}) = \frac14(S_0^2-\|\mathbf{S}\|^2)$	Degree of polarization: $P = \|\mathbf{S}\|/S_0$
cone $N(\tilde{\rho}) = 0$	Fully polarized states = Poincaré sphere
$r_e = e^2/(4\pi\epsilon_0 mc^2)$	Classical electron radius
$d\sigma_T/d\Omega = r_e^2\|\hat{\varepsilon}'\cdot\hat{\varepsilon}\|^2$	Thomson differential cross-section
$\sigma_T = 8\pi r_e^2/3$	Thomson total cross-section; $0.66524587$ barn
Mueller matrix (scattering-plane basis)	$d\sigma_T/d\Omega$ as the linear map on $(I,Q,U,V)$; $Q$ positive for polarization perpendicular to the scattering plane
$P(\theta) = \sin^2\theta/(1+\cos^2\theta)$	Degree of linear polarization of unpolarized Thomson-scattered light
$P_c(\theta) = 2\cos\theta/(1+\cos^2\theta)$	Degree of circular polarization retained by a circularly polarized beam
$\omega' = \omega/[1+x(1-\cos\theta)]$	Compton shift
$x = \hbar\omega/mc^2$	The one dimensionless scale
$d\sigma_{\mathrm{KN}}/d\Omega$	Klein–Nishina differential cross-section
$\sigma_{\mathrm{KN}}(x)\to\sigma_T$	Thomson limit
$A(\omega,\theta)+B(\omega,\theta)\cos2\phi$	Azimuthal modulation for a linearly polarized incident beam; $\|B/A\|$ is the analysing power
$\mathbf{S}_e\cdot(\hat{\mathbf{k}}\times\hat{\mathbf{k}}')$	Electron-spin asymmetry (magnetic Compton)
$\tilde{\rho}_e = \frac12(e_0+i\mathbf{s}\cdot\tilde{e})\in\mathbb{M}_+$	Electron spin biquaternion; $N(\tilde{\rho}_e) = \frac14(1-\|\mathbf{s}\|^2)$