Photon plane coupling

Summary: Restricting a magnetic two-form overlap to the photon's actual two-plane gives a rank-two projector and an explicit polarization-dependent coupling diagnostic.

Spatial setting and conditional interaction

All directions below are ordinary macroscopic spatial directions in one nested physical manifold. Work in a flat tangent chart of a containing sector, with real Euclidean space $V$ of dimension $d\geq3$. The photon has an oriented physical two-plane $E\subset V$ and unit phase direction $k\in E^\perp$. Its complex polarization belongs to $E\otimes\mathbb C$. This is the geometry in photon transversality.

The calculation adopts the spatial two-form pairing of the dimensional magnetic kernel. It is one specified interaction term of the single field $\Psi_\infty$. It does not introduce mediator particles or establish that this term is the whole electromagnetic response. Electric, orbital, and additional spatial response terms must be retained when the action supplies them.

For skew matrices use the Hermitian pairing and norm

$$\langle A,B\rangle=\frac12\operatorname{tr}(A^\dagger B), \qquad \|A\|^2=\langle A,A\rangle.$$

For real vectors define $u\wedge v=uv^T-vu^T$. A unit linear polarization $\varepsilon\in E$ gives the normalized wave two-form

$$B(\varepsilon)=\varepsilon\wedge k, \qquad \|B(\varepsilon)\|^2=1. \tag{1}$$

Its two-form plane includes $k$; that is a description of the wave's spatial curvature. The photon's bound oscillation plane remains $E$.

Restrict the field projector to the photon plane

Choose real orthonormal $\varepsilon_1,\varepsilon_2$ spanning $E$, write $P_E=\sum_a\varepsilon_a\varepsilon_a^T$, and set $B_a=\varepsilon_a\wedge k$. The two matrices $B_a$ are orthonormal. For any complex skew matrix $M$,

$$\langle B_a,M\rangle=\varepsilon_a^T M k.$$

The orthogonal projector onto these two admissible wave amplitudes is

$$\boxed{ \mathcal Q_{E,k}M =\sum_{a=1}^2 B_a\langle B_a,M\rangle =w\wedge k,\qquad w=P_E M k.} \tag{2}$$

Because $w\perp k$, applying $\mathcal Q_{E,k}$ again leaves it unchanged. The orthonormal expansion also proves self-adjointness, positivity, and rank two. In particular,

$$\boxed{\|\mathcal Q_{E,k}M\|^2=\|P_E M k\|^2.} \tag{3}$$

The earlier closed-field projector is

$$\mathcal P_kM=P_kM+MP_k, \qquad P_k=kk^T.$$

It admits every polarization in $k^\perp$ and has rank $d-1$. Our photon has only the two amplitudes selected by $E$. Consequently

$$\mathcal Q_{E,k}\mathcal P_k =\mathcal P_k\mathcal Q_{E,k}=\mathcal Q_{E,k}, \qquad 0\leq\mathcal Q_{E,k}\leq\mathcal P_k. \tag{4}$$

These projectors coincide at depth three, where $E=k^\perp$. At larger depths a sum over all $d-1$ transverse amplitudes is a different diagnostic from a sum over this photon's two polarizations.

Definite polarization and the sum over two polarizations

For one real spatial rotation tensor $J$, define

$$h_a=\varepsilon_a^T Jk,\qquad C(E,k;J)=h^\dagger h=\|P_EJk\|^2. \tag{5}$$

A normalized complex polarization $c\in\mathbb C^2$ has wave two-form $B(c)=\sum_a c_aB_a$. The squared overlap is

$$|\langle B(c),J\rangle|^2 =|c^\dagger h|^2 =c^\dagger hh^\dagger c. \tag{6}$$

Thus the maximum over photon polarizations is $C$. When $h\ne0$, one orthogonal complex polarization has zero overlap with this one tensor. For real $J$, the two circular polarizations each give $C/2$. An unpolarized ensemble, described by $I_2/2$, also gives $C/2$; the sum over the two orthonormal input polarizations is $C$, not that average.

The overlap vanishes for every photon polarization in this plane exactly when $P_EJk=0$. Shared spatial directions alone do not decide that equation. For a normalized complex structure $J^2=-I$, $|Jk|=1$ still permits $P_EJk=0$ when the containing geometry allows $E$ perpendicular to $Jk$. The earlier full-transverse value one therefore does not imply that every photon preparation has nonzero overlap.

Several target responses

If target response $\lambda$ has tensor $M_\lambda$, use $h_{\lambda a}=\langle B_a,M_\lambda\rangle$. For independently distinguishable output responses with nonnegative weights $w_\lambda$, the input-polarization matrix is

$$W=\sum_\lambda w_\lambda h_\lambda h_\lambda^\dagger, \qquad c^\dagger Wc\geq0. \tag{7}$$

A dark input must be orthogonal to every response with positive weight. Two independent response vectors can make $W$ positive definite, removing the dark polarization present for a single tensor. Contributions to the same coherent output must instead be added as amplitudes before squaring.

Equations (1)–(7) are angular and algebraic results. A physical transition amplitude additionally integrates spatial profiles and phases and uses the action's mode normalization. A probability or rate requires temporal dynamics, final-state availability, and incident-flux normalization.

Verification and related

Polarized photon transition strength extends the frozen source calculation to every target polarization tangent in the adopted spin module, including helicity-sensitive response zeros.

polarization_coupling_checks.py checks the projector by an independently assembled orthonormal two-form basis, tests complex inputs, and compares direct matrix pairings with polarization amplitudes in rotated spatial frames.

• Dimensional magnetic kernel — supplies the assumed two-form term and full-transverse projector.
• Generalized sector coupling — places the overlap within reciprocal, normalized dynamics of the same field.

Bilateral polarization coupling retains both objects' polarization and response. A zero static source overlap does not by itself make the mixed response derivative vanish.