Polarized photon transition strength

Summary: Summing all spinor response directions gives an exact two-polarization transition matrix for a simple spatial Pauli probe, including helicity-sensitive zeros and a nonzero total-strength bound at every tested depth.

Include the target's allowed polarization changes

The photon-plane calculation compares a fixed target source with a specified photon. An incident wave also perturbs the target. This note computes that response for a particular spatial Pauli term, keeping all polarization variations in the adopted spin module.

Work in a finite Euclidean spatial chart of the one nested macroscopic manifold. Let $k,e_1,e_2$ be orthonormal ordinary spatial vectors, with $e_1,e_2$ the photon plane and $k$ a propagation direction perpendicular to it. At higher depth these are three selected directions within the containing space, not a separate three-dimensional compartment.

Let $z$ be a normalized target spinor. In even dimensions use a selected half-spin representation; the following bivector operators preserve it. Their components describe the spatial field's polarization and do not add physical dimensions. This is a frozen-envelope Pauli benchmark, not the complete electromagnetic current or a derived neutrino coupling.

The two spatial probe operators

With $\gamma(v)=v^a\gamma_a$ and Euclidean Clifford relations, define

$$T_a=-i\gamma(e_a)\gamma(k),\quad a=1,2, \qquad T_0=-i\gamma(e_1)\gamma(e_2). \tag{1}$$

All three are Hermitian and square to the identity. Direct multiplication using the three orthogonal vectors gives

$$T_1T_2=iT_0,\qquad T_2T_1=-iT_0. \tag{2}$$

A definite complex photon polarization $c=(c_1,c_2)^T$, $c^\dagger c=1$, acts through $T(c)=c_1T_1+c_2T_2$. The actual coupling prefactor, spatial integral and time dependence must still be obtained from the action.

Define the three real target polarization components

$$h_a=z^\dagger T_a z,\qquad s=z^\dagger T_0z, \qquad C=h_1^2+h_2^2. \tag{3}$$

For an orthonormal basis $u_\lambda$ of the complete complement $z^\perp$, the transition amplitudes are $u_\lambda^\dagger T(c)z$. The source expectation and the orthogonal response are different quantities.

Exact transition matrix

Completeness of the response basis gives

$$G_{ab}=\sum_\lambda \big(u_\lambda^\dagger T_a z\big)^* \big(u_\lambda^\dagger T_b z\big) =z^\dagger T_a(I-zz^\dagger)T_bz. \tag{4}$$

Using (2), this becomes

$$\boxed{G= \begin{pmatrix} 1-h_1^2&-h_1h_2+is\\ -h_1h_2-is&1-h_2^2 \end{pmatrix}.} \tag{5}$$

Thus the complete squared response in these normalized amplitude units is

$$\sum_\lambda|u_\lambda^\dagger T(c)z|^2=c^\dagger Gc. \tag{6}$$

Its trace, determinant and eigenvalues are

$$\operatorname{tr}G=2-C,\qquad \det G=1-C-s^2, \qquad g_\pm=1-\frac C2\pm\sqrt{s^2+\frac{C^2}{4}}. \tag{7}$$

Positivity implies $C+s^2\leq1$. Equivalently, these are the three expectations of a Pauli algebra and form a Bloch vector of length at most one. A complex input polarization is dark precisely when it lies in $\ker G$. Such a kernel exists exactly when $C+s^2=1$.

For a real linear polarization $c$, the imaginary antisymmetric part drops out and (6) is $1-(c^Th)^2$. For circular polarizations,

$$c_\pm=\frac{(1,\pm i)^T}{\sqrt2},\qquad \boxed{c_\pm^\dagger Gc_\pm=1\mp s-\frac C2.} \tag{8}$$

The target's polarization therefore distinguishes the two helicity responses. Averaging the two incident polarizations yields $1-C/2$; it is not the squared unperturbed source expectation $C/2$.

A full response cannot be dark to both polarizations

Since $C\leq1$, equation (7) gives the dimension-independent bound

$$\boxed{\operatorname{tr}G\geq1.} \tag{9}$$

Consequently this nonzero simple Pauli coupling cannot annihilate the complete two-polarization response on any normalized pure target spinor. One circular or linear polarization may be blocked; both cannot be blocked simultaneously in the complete module. If $h_1=h_2=0$, the unperturbed source vanishes but the total transition strength is instead two.

For a real polarization operator $T$ with $T^2=I$, a finite pulse gives an independent exact check:

$$e^{-i\theta T}z=(\cos\theta)z-i(\sin\theta)Tz, \qquad 1-|z^\dagger e^{-i\theta T}z|^2 =\sin^2\theta\big[1-(z^\dagger Tz)^2\big]. \tag{10}$$

This measures departure from the original ray. A nonzero source expectation can still produce an elastic phase response, so (9)–(10) are statements about the orthogonal transition component, not a partition of every possible scattering effect.

Five- and six-dimensional isotropic comparison

Spin$(5)$ and either Spin$(6)$ half-spin module have four complex components. On both, the subgroup generated by the three operators in (1) consists of two equivalent spin-one-half copies. One can identify the module for this calculation with $\mathbb C^2\otimes\mathbb C^2$ so that these probes act on the first factor. This tensor notation is a representation decomposition, not a new physical spatial factorization.

Uniform spatial rotations of a fixed pure ray give the same invariant probability distribution on $\mathbb{CP}^3$ in both cases. Spin$(5)$ is transitive there, and its invariant distribution agrees with the Fubini–Study distribution, which is also invariant under Spin$(6)$. The prior isotropic comparison provides the underlying spatial ensemble and its assumptions.

Therefore the entire joint distribution of $(h_1,h_2,s)$, and consequently of $G$, is identical under this specified ensemble. In particular

$$\mathbb E\,h_a^2=\frac15,\qquad \mathbb E\,G=\frac45I_2, \qquad \mathbb E\,(c^\dagger Gc)=\frac45 \tag{11}$$

for any fixed normalized complex photon polarization. The moment follows directly from the normalized pure-state identity $\mathbb E(zz^\dagger)^{\otimes2}=(I+\mathrm{Swap})/[4\cdot5]$: unitary invariance makes the average proportional to the symmetric projector, whose trace is ten; normalization fixes the coefficient. Each $T_a$ is traceless with $\operatorname{tr}T_a^2=4$. Correlated spatial preparations and actual mode frequencies must be calculated rather than averaged away.

What can still distinguish physical couplings

Equation (6) sums unweighted polarization transitions in a fixed profile. It does not impose the energy of the incident photon, include all spatial profile variations, or normalize the incident flux. The action can supply different spatial form factors, kinetic coefficients and accessible frequency spectra. The polarized response spectrum retains those frequency and exposure restrictions.

A coupling generated by a higher spatial operation need not reduce to the simple bivectors in (1). For example, rotation-neutral modes cancel a sum of two spatial rotation generators as an operator; (9) does not apply to that sum. Identifying the actual higher-sector source is therefore substantive. The photon having two bound spatial dimensions does not alone select (1) as the complete interaction law.

Verification

spatial_coupling_mechanism_checks.py constructs independent Clifford representations at depths three, four, five, six and ten. It sums the full response complement, checks (5)–(9) for complex polarized targets, and compares (10) with matrix-exponential evolution. These test the conditional operator calculation, not physical particle rates.

Related

• Photon plane coupling — specifies the two transverse spatial bivectors and the unperturbed source overlap.
• Isotropic polarization comparison — proves the fixed-source distribution used as the starting comparison.
• Bilateral polarization coupling — explains why target variations must accompany the incident polarization.