Isotropic polarization comparison

Summary: Uniform five-dimensional null-axis and six-dimensional complex-structure tensors have identical photon-plane overlap distributions under an explicitly isotropic preparation.

The preparation is part of the model

Use the two-plane diagnostic from photon plane coupling:

$$C_d=\|P_EJ_dk\|^2, \qquad |k|=1,\qquad E\subset k^\perp,\qquad \dim E=2. \tag{1}$$

The real skew tensors are the fixed, normalized candidates already used in the dimensional magnetic kernel:

$$J_6^2=-I_6, \qquad J_5^2=-I_5+nn^T,\qquad |n|=1. \tag{2}$$

Choose $k$ uniformly on the sphere in each containing spatial space. Conditional on $k$, choose $E$ uniformly among two-planes in $k^\perp$. This is a mathematical ensemble of relative preparations. It does not follow from nesting, transversality, or the particle's depth. An actual observer, binding geometry, or preparation may restrict this ensemble.

Both objects are polarized in each preparation. Fixing $J_d$ chooses the target's frame; it does not replace its polarization by an average. Independent isotropic rotations of both objects give the same relative ensemble. Correlations between their polarizations would change it.

All coordinates remain ordinary spatial directions of one manifold. Equation (2) describes a specified uniform rotational response tensor; it does not establish the full shape or field solution of either particle.

Six dimensions

For every $k$, $J_6k$ is a unit vector perpendicular to $k$. The space $k^\perp$ has dimension five. Its squared projection onto a uniformly oriented two-plane has the beta law

$$\boxed{C_6\sim\operatorname{Beta}\left(1,\frac32\right).} \tag{3}$$

To obtain this law, rotate the plane to the first two axes and express a uniform direction as a vector of five independent standard Gaussians divided by its norm. The numerator is the sum of two squared Gaussians; the remaining three supply the complementary denominator. The gamma ratio derivation and normalization are given in full in isotropic projection fractions.

Five dimensions

First put

$$A=|J_5k|^2=1-(n\cdot k)^2.$$

The four directions perpendicular to $n$ give $A\sim\operatorname{Beta}(2,1/2)$. Conditional on $k$, the normalized direction $J_5k/|J_5k|$ lies in a four-dimensional $k^\perp$. Its squared projection onto a uniform two-plane is

$$B\sim\operatorname{Beta}(1,1),\qquad C_5=AB. \tag{4}$$

The conditional law of $B$ is the same for every non-null $k$, so $A$ and $B$ are independent. The exactly null directions have sphere measure zero and do not change this distribution. The density of $A$ is

$$f_A(a)=\frac34\,a(1-a)^{-1/2},\qquad 0<a<1.$$

Since $B$ is uniform, the product density is

$$f_{C_5}(c) =\int_c^1\frac{f_A(a)}{a}\,da =\frac32\sqrt{1-c}. \tag{5}$$

This is precisely the density in (3). Consequently

$$\boxed{C_5\overset{\rm law}=C_6 \sim\operatorname{Beta}\left(1,\frac32\right).} \tag{6}$$

The five-dimensional tensor's null axis lowers $|J_5k|$, while projection into the photon plane takes place in a smaller transverse space. Under this ensemble the two effects give the same complete distribution.

Exact moments and small-overlap fraction

For either depth,

$$\mathbb E[C]=\frac25, \qquad \operatorname{Var}(C)=\frac{12}{175}, \qquad \Pr(C\leq t)=1-(1-t)^{3/2},\quad 0\leq t\leq1. \tag{7}$$

More generally, $\mathbb E[C^r]=(1)_r/(5/2)_r$ for $r>-1$. Near zero the fraction is $3t/2+O(t^2)$. For example, about $1.50\%$ of this ensemble has $C\leq0.01$, at both depths. This threshold refers to a normalized angular overlap, not a scattering probability.

The normalization in (2) fixes each nonzero rotational singular value to one. It does not fix the Frobenius norm to be equal: $\|J_5\|^2=2$ and $\|J_6\|^2=3$. Renormalizing the tensors or using action-derived physical coefficients changes the scale and must be declared. No physical charge or coupling coefficient is inferred from this comparison.

A general adjacent-dimension identity

Let $J_{2m}$ be an orthogonal complex structure and let $J_{2m-1}$ have one unit null axis and unit rotational singular values on its complement. Choose a $p$-plane $E\subset k^\perp$ by the same ensemble, with $1\leq p\leq2m-2$. Then both depths give

$$\boxed{\|P_EJ_dk\|^2 \sim\operatorname{Beta}\left(\frac p2, \frac{2m-1-p}{2}\right),\qquad d=2m-1,2m.} \tag{8}$$

For the odd depth the factors have beta parameters $(m-1,1/2)$ and $(p/2,m-1-p/2)$. To prove the product rule, take independent unit-scale gamma variables $X,Y,Z$ with shapes $p/2,m-1-p/2,1/2$. The ratios $X/(X+Y)$ and $(X+Y)/(X+Y+Z)$ are independent; their product is $X/(X+Y+Z)$, with exactly the beta parameters in (8). If the middle shape is zero, the first ratio is the constant one and the result follows directly. These tensor assumptions are not asserted for every spinor or every physical particle at the indicated depths.

A definite photon polarization

Let $h=(\varepsilon_1^TJ_dk,\varepsilon_2^TJ_dk)$ in an orthonormal frame of the photon plane. The relative ensemble is invariant under rotating that frame, so $h$ has a uniform angular direction conditional on $|h|^2=C_d$. Equation (6) therefore gives the same complete real two-component distribution of $h$ at depths five and six.

For any fixed normalized complex polarization $c\in\mathbb C^2$, the intensity $I_c=|c^\dagger h|^2$ consequently has the same distribution at both depths. This statement includes each object's definite polarization; it does not require averaging the photon over its two states. Its first two moments are

$$\boxed{\mathbb E[I_c]=\frac15, \qquad \operatorname{Var}(I_c) =\frac{3+5|c^Tc|^2}{175}.} \tag{9}$$

To derive the second moment, write $I_c=h^TR_ch$ with $R_c=\operatorname{Re}(cc^\dagger)$. The radial law and uniform angle give $\mathbb E[h_i h_j]=\delta_{ij}/5$ and

$$\mathbb E[h_i h_j h_k h_l] =\frac{\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl} +\delta_{il}\delta_{jk}}{35}.$$

Since $\operatorname{tr}R_c=1$ and $\operatorname{tr}(R_c^2)=(1+|c^Tc|^2)/2$, equation (9) follows. For linear polarization, $I_c\sim\operatorname{Beta}(1/2,2)$ and $\Pr(I_c\leq t)=3\sqrt t/2-t^{3/2}/2$. For circular polarization, $I_c=C_d/2$. Ellipticity changes the intensity distribution while preserving the equality between the two specified dimensional models.

Interpretation

This exact equality identifies which information a uniform isotropic tensor model retains. To obtain a distinctive five-dimensional response, the calculation needs additional derived spatial information: a restricted relative preparation, spatially varying response patterns, the allowed transition spectrum, or other terms of the same field's action. Those structures can be tested before any orientation average is taken.

Verification

polarization_coupling_checks.py checks beta product densities by independent quadrature, moments and CDFs, and separately samples orthonormal spatial frames. The probability law follows from the derivation; finite samples are a supplementary check.

Related

• Photon plane coupling — defines the rank-two diagnostic and distinguishes a sum from an unpolarized average.
• Isotropic projection fractions — derives the underlying probability law from spatial Gaussian coordinates.
• Dimensional magnetic kernel — supplies the specific uniform five- and six-dimensional tensors.