Complete spatial polarization data

Summary: Complete spatial Clifford tensors retain the amplitudes and phases of a polarized field, exposing information that a circulation tensor captures at six dimensions but can omit at ten.

Begin with the full oscillating field

For a monochromatic vector with complex amplitude $a+ib$, where $a,b$ are real spatial vectors,

$$E(t)=\operatorname{Re}[(a+ib)e^{-i\omega t}] =a\cos\omega t+b\sin\omega t. \tag{1}$$

Its trajectory lies in $\operatorname{span}_{\mathbb R}\{a,b\}$, even when the containing physical space has many dimensions. The squared ellipse axes are the eigenvalues of

$$\begin{pmatrix}|a|^2&a\cdot b\\a\cdot b&|b|^2\end{pmatrix}. \tag{2}$$

Parallel $a,b$ give linear polarization; orthogonal equal-length $a,b$ give circular polarization. More spatial dimensions permit more plane orientations. A field with several modes, spatially varying amplitudes or different frequencies has further structure which (1) does not describe. The Dirac polarization waves give an exact multimode example with a shared scalar source and changing spatial currents.

For a particle field, the allowed components must come from its actual spatial equation. In a fixed degenerate spatial multiplet, write

$$\Psi_z(x,t)=e^{-iMt}W(x)z,\qquad \int W^\dagger W\,dx=I_N,\qquad z^\dagger z=1. \tag{3}$$

The spatial Dirac construction supplies examples where $W(x)$ includes both radial components and all bound coordinates. The coefficients $z$ describe the polarization of this field. Their number is not the number of physical spatial dimensions. All $x$ coordinates are ordinary macroscopic directions in the one nested manifold.

Equation (3) is a fixed-multiplet description. It does not assert that every physical particle has that profile, or that its other spatial modes can be discarded.

Retain amplitudes and relative phases without selecting one tensor

The matrix

$$\varrho=zz^\dagger,\qquad \varrho^\dagger=\varrho,\quad \varrho^2=\varrho,\quad \operatorname{tr}\varrho=1 \tag{4}$$

retains every relative amplitude and phase in $z$. It removes only the common constant phase. For any operator $O$ on the multiplet,

$$\langle O\rangle=\operatorname{tr}(\varrho O). \tag{5}$$

A common phase relative to another coherent field must still be retained in the combined field and its transition terms. Likewise, replacing $z$ by a position-dependent phase cannot be justified by (4); its derivatives enter the spatial equation, as shown in Spatial polarization textures.

To express (4) as spatial tensors, choose Hermitian Euclidean Clifford matrices $\gamma_i$, satisfying $\{\gamma_i,\gamma_j\}=2\delta_{ij}I$. They describe how the field transforms under rotations of its actual spatial coordinates. For even depth use one specified chiral representation; for odd depth use the stated irreducible spin representation.

Define the Hermitian spatial operators

$$B_{ij}=-i\gamma_i\gamma_j,\qquad C_{ijkl}=\gamma_i\gamma_j\gamma_k\gamma_l, \qquad i<j<k<l. \tag{6}$$

Restriction to the selected chiral space is understood. Their expectation values are the two-form $J_{ij}=\operatorname{tr}(\varrho B_{ij})$ and four-form $K_{ijkl}=\operatorname{tr}(\varrho C_{ijkl})$.

Count and verify a complete operator basis

At the depths below, the listed operators $\Gamma_A$ form a complete Hermitian basis on the indicated polarization multiplet:

In five dimensions a four-form is dual to a vector in those same five spatial directions. The table counts descriptions of fields, not additional spatial coordinates or independent particles.

The Clifford trace identities give

$$\operatorname{tr}(\Gamma_A\Gamma_B)=N\delta_{AB}. \tag{7}$$

To see why the stated grades suffice, products of distinct Clifford generators are trace-orthogonal unless their index sets agree or are related by the top-dimensional duality. In an irreducible odd-dimensional module, the top product is scalar; on an even-dimensional chiral module it is fixed by chirality. The table selects one representative of each dual pair. Its $N^2$ orthogonal operators span the entire $N\times N$ matrix algebra.

Consequently

$$\boxed{\varrho=\frac1N\sum_A \operatorname{tr}(\varrho\Gamma_A)\Gamma_A.} \tag{8}$$

The scalar coefficient is one. For a single normalized coherent polarization, purity also gives

$$\sum_{A\ne0}\operatorname{tr}(\varrho\Gamma_A)^2=N-1. \tag{9}$$

The coefficients are constrained by the full equation $\varrho^2=\varrho$; (9) alone is insufficient. Mixtures obey (8) too, but must satisfy positivity and have $\operatorname{tr}\varrho^2\le1$.

At six dimensions, $J$ therefore determines the entire polarization matrix within the selected four-component multiplet:

$$\boxed{\varrho_6=\frac14\left(I+\sum_{i<j}J_{ij}B_{ij}\right).} \tag{10}$$

For a pure ray, its $J$ satisfies $J^2=-I_6$. One way to establish this for every ray is to note that the fifteen traceless $B_{ij}$ span the Hermitian generators of $SU(4)$. Their spin representation acts transitively on the rays, and rotates $J$ as a physical two-form. A reference ray has three unit orthogonal planes; rotating it preserves $J^2=-I_6$. Arbitrary complex coefficients in this fixed chiral multiplet already include all its constant polarizations. Additional six-dimensional complexity must also consider other spatial modes and textures.

At ten dimensions the complete expression is instead

$$\boxed{\varrho_{10}=\frac1{16} \left(I+\sum_{i<j}J_{ij}B_{ij} +\sum_{i<j<k<l}K_{ijkl}C_{ijkl}\right).} \tag{11}$$

The circulation tensor alone can lose relative-phase information.

An explicit ten-dimensional coherent family

Construct five fermionic matrix pairs $a_j=(\gamma_{2j-1}+i\gamma_{2j})/2$. Let $|0\rangle$ be their common vacuum and $|1234\rangle=a_1^\dagger a_2^\dagger a_3^\dagger a_4^\dagger|0\rangle$, with its phase fixed so that $\langle0|C_{1357}|1234\rangle=1$. These are basis labels for components of one spatial field, not constituent particles or extra coordinates.

The normalized same-chirality family

$$\boxed{z_{\eta,\theta} =\cos\eta\,|0\rangle+e^{i\theta}\sin\eta\,|1234\rangle} \tag{12}$$

has

$$\boxed{J=\operatorname{diag} (\lambda J_2,\lambda J_2,\lambda J_2,\lambda J_2,J_2), \quad \lambda=\cos2\eta,\quad J_2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.} \tag{13}$$

Every off-diagonal occupation transition in a two-form changes at most two matrix occupations, so it cannot connect the two components in (12). The diagonal two-forms are $1-2a_j^\dagger a_j$, giving (13). The four-form does connect them:

$$\boxed{K_{1357}=\sin2\eta\cos\theta.} \tag{14}$$

Thus $\theta$ can change the field polarization without changing any component of $J$. At $\eta=\pi/4$, the fields with $\theta=0,\pi$ are orthogonal, yet have exactly the same $J$.

Every vector in (12) is a pure coherent state in the sense of (4). The term Cartan-pure spinor imposes a separate algebraic condition: the complex spatial vectors $v$ satisfying $\gamma(v)z=0$ span a maximal five-dimensional complex annihilator. The vacuum has that property; for $0<\eta\le\pi/4$, (12) has only a one-dimensional complex annihilator. Indeed, write a general complex Clifford vector as $\sum_j(u_j a_j+v_j a_j^\dagger)$. Acting on (12), its one-occupation terms force every $v_j=0$, while its three-occupation terms force $u_1=\cdots=u_4=0$. Both coefficients in (12) are nonzero in this range, and the distinct occupation grades cannot cancel each other. Only $a_5$ remains. On the vacuum, all five $a_j$ annihilate the field coefficient.

This complex algebraic test concerns combinations of the existing spatial directions. It does not remove real physical directions from the particle. The Clifford construction and distinction between pure and general spinors provide a mathematical comparison; (7)–(14) are checked directly here.

Even a two-form interaction can read the additional information

Let a specified complex spatial rotation pattern act through $O_f=\sum_A f_A B_A$, where $A$ labels coordinate planes. For a pure polarization, its component orthogonal to the initial mode has squared norm

$$\boxed{\|(I-\varrho)O_fz\|^2=f^\dagger\mathcal V f,\qquad \mathcal V_{AB}=z^\dagger B_AB_Bz-J_AJ_B.} \tag{15}$$

This is a positive response matrix. It retains relative phases in the incoming pattern $f$ as well as the occupied field $z$.

For disjoint ordered planes, $B_{ij}B_{kl}=-C_{ijkl}$. Shared indices instead reduce to a two-form, and identical planes give the identity. Consequently the response to a two-form probe can depend on four-form polarization data even though its mean source depends only on $J$. No additional four-form interaction term is required for this distinction.

Equal-current spatial polarizations constructs the full bound fields and a reciprocal six-inside-ten example. The periodic polarization test supplies the corresponding approach when complete field resolvents and sidebands are needed. Neither an instantaneous response norm nor this operator classification is a physical scattering rate.

Verification

spatial_polarization_structure_checks.py constructs all the actual matrices and verifies their full trace Gram matrices, reconstruction of arbitrary complex rays and mixtures, the ten-dimensional phase family and its Clifford annihilators. Independent operator products check the response in (15). Reports are timestamped and checks remain active under Python optimization.