Spatial Dirac polarization waves

Summary: The full spatial Dirac field supports exact linear polarization waves and nonlinear polarized plane superpositions whose scalar binding source stays fixed while their spatial currents change.

Restore the polarizations omitted by a single-amplitude envelope

The scalar envelope fixes a rest polarization while following changes of the bound profile. The scalar spatial source itself leaves a family of rest polarizations degenerate. Their normal propagation can be derived from the full spatial Dirac equation, without assuming that those motions are frozen or gapped.

All fields below occupy the same ordinary bound spatial coordinates $x\in\mathbb R^d$ and an existing normal coordinate $y$ in a containing sector. The polarization amplitudes specify spatial field patterns on that manifold. Their component labels do not add physical dimensions.

Choose an orthonormal set of allowed rest spinors collected in a matrix $V$, with $V^\dagger V=I_N$ and $\beta V=V$. Every column belongs to the containing spatial Clifford representation. A further selected chirality can be imposed when it defines the intended radial family, but scalar binding alone does not select that chirality or the occupied polarization.

A full spatial frame for the polarization amplitudes

For a normalized real radial profile $f,g$, define

$$F(x)=\frac{[f(r)I+i g(r)\alpha(\widehat x)]V} {\sqrt{\Omega_{d-1}}},\qquad W(x)=\big(F(x),\alpha_yF(x)\big). \tag{1}$$

The bound equation and its normal partner satisfy

$$H_bF=MF,\qquad H_b\alpha_yF=-M\alpha_yF, \quad H_b=-i\alpha_i\partial_i+\beta\mathfrak m(x), \tag{2}$$

with $M>0$. Thus, after integration over the actual bound coordinates,

$$W^\dagger W=I_{2N},\qquad \alpha_yW=W(\sigma_1\otimes I_N),\qquad H_bW=W(M\sigma_3\otimes I_N). \tag{3}$$

The normalized spinor amplitudes therefore retain the normal kinetic operator. They have not been obtained by dropping its negative-frequency partner.

The scalar identity is stronger: it holds pointwise,

$$\boxed{W(x)^\dagger\beta W(x) =\sigma_0(x)(\sigma_3\otimes I_N),\qquad \sigma_0(x)=\frac{f(r)^2-g(r)^2}{\Omega_{d-1}}.} \tag{4}$$

To check it, the diagonal blocks follow from $\beta V=V$ and anticommutation with $\alpha(\widehat x)$. The mixed block vanishes: $F^\dagger\beta\alpha_yF=0$. Its radial cross terms cancel because $\beta\alpha_y$ commutes with $\alpha(\widehat x)$, and the remaining terms connect opposite $\beta$ eigenspaces.

For any complex amplitude $\chi\in\mathbb C^{2N}$, the reconstructed field has

$$\Psi=W\chi,\qquad \overline\Psi\Psi=\sigma_0(x)s, \qquad s=\chi^\dagger(\sigma_3\otimes I_N)\chi. \tag{5}$$

Its scalar source depends on the total proper scalar density, while its full spatial current can distinguish the polarizations and their phases.

Linear polarization waves are full-field modes

Let the rest background be $\chi_0=\sqrt{s_0}(z,0)^Te^{-iMt}$, with $z^\dagger z=1$. Readjust the radial profile to its scalar density $s_0$ as usual. Take a perturbation in any rest polarization $z_\perp$ orthogonal to $z$. Equation (4) gives

$$\delta(\overline\Psi\Psi)=0 \tag{6}$$

at every bound spatial point, for either component of its normal partner. The scalar center and all scalar moments consequently have zero first variation too. The linearized full field equation in this direction has no additional source-feedback term.

In the frame rotating at the background phase frequency, the two normal amplitudes $\eta$ obey

$$\boxed{i\partial_t\eta =\big[-i\sigma_1\partial_y+M\sigma_3-MI_2\big]\eta.} \tag{7}$$

These are exact linearized modes of the specified full spatial source model. Their physical normal wavenumber $k$ gives

$$\boxed{\omega_\pm(k)=-M\pm\sqrt{M^2+k^2}.} \tag{8}$$

In particular,

$$\omega_+(k)=\frac{k^2}{2M}+O(k^4/M^3). \tag{9}$$

There is no positive gap isolating one chosen rest polarization from its orthogonal neighbors. At $k=0$, this branch is the tangent to the stationary polarization family. Its long-wave frequency is quadratic, which must be retained if these modes are included in a coupling reduction. It is not a derived photon dispersion relation.

The radial source can have separate unstable modes; (8) does not establish stability of the entire background. It identifies a specific set of polarization motions that a fixed-polarization envelope omits.

Exact finite polarized waves can share one unchanged source

The same pointwise identity also gives nonlinear solutions beyond first order. Choose mutually orthonormal rest polarizations $z_j$ and positive proper densities $s_j$ with $\sum_j s_j=s_0$. Solve the radial source once at $s_0$, obtaining $M$. For arbitrary normal momenta $k_j$, put

$$E_j=\sqrt{M^2+k_j^2},\qquad u_j=\begin{pmatrix} \sqrt{(E_j+M)/(2E_j)}\\ k_j/\sqrt{2E_j(E_j+M)} \end{pmatrix}. \tag{10}$$

The amplitude field

$$\boxed{\chi(t,y)= \sum_j\sqrt{\frac{s_jE_j}{M}}\, (u_j\otimes z_j)e^{i(k_jy-E_jt)}} \tag{11}$$

has constant scalar density $s_0$. Indeed, $u_j^\dagger\sigma_3u_j=M/E_j$, and every cross term in (5) vanishes by $z_i^\dagger z_j=0$ for $i\ne j$. Each term solves the same frozen normal Dirac equation, and their sum preserves precisely the scalar source which made that operator self-consistent. Therefore $\Psi=W\chi$ solves the full nonlinear scalar source equation exactly.

The laboratory number density and normal current, integrated over the bound coordinates, are

$$\boxed{n=\sum_j\frac{s_jE_j}{M},\qquad j_y=\sum_j\frac{s_jk_j}{M}.} \tag{12}$$

The relativistic normalization factors in (11) are needed for both the source and the current. Replacing them by unit laboratory amplitudes would generally change the scalar source and require another profile.

These uniform waves are modes of the same field sharing one bound profile. This construction does not replace the distinct-depth, mutually deformed pair problem with two independently assigned particles.

Scalar density does not describe the complete spatial motion

Orthogonal polarizations remove the scalar cross terms in (5). They need not remove cross terms in $\Psi^\dagger\alpha_i\Psi$. Distinct $E_j,k_j$ can therefore produce a changing bound spatial current even while every scalar binding moment remains constant.

The diagnostic uses $s_1=0.7$, $s_2=0.3$, $k_1=0.4M$ and $k_2=-0.7M$ with two complex orthogonal spatial polarizations. At depths three, five and six, it verifies the unchanged scalar source and an explicitly changing local bound current. Both full spatial polarizations are needed to describe that solution; the scalar density alone loses the motion.

For a varying total $s(t,y)$, the bound profile also varies. Replacing $W$ by $W_s$ restores the profile-derivative residual calculated in the scalar-envelope note, now with $\sigma_1\otimes I_N$. Constant $s$ makes that residual vanish, including (11). General wavepackets require the actual radial, shape and center response as well.

This gives a precise coupling limit: scalar self-binding permits and transports different polarizations, but does not choose the occupied one. A physical neutrality mechanism must supply that selection and the current sampled by an incoming field; it cannot infer them from the unchanged scalar source alone.

Reflection symmetry of scalar Dirac sources shows why this degeneracy can persist with unequal scalar confinement axes. It also checks the nonlinear moment source before treating a frozen eigenspace as a degenerate family of self-consistent fields.

Two-frequency spatial Dirac currents uses opposite-frequency components of this same frame to construct an exact current oscillating at twice the bound frequency. Its unit-norm family separates a slow clock from a small current or a small source response.

Complete spatial polarization data identifies which tensors retain all constant amplitudes and phases in the chosen multiplet. Spatial polarization textures derives the additional kinetic terms when those amplitudes vary across the bound coordinates, including the component outside the retained frame. The source-preserving normal waves above and bound-coordinate textures therefore require their respective complete spatial equations.

Verification

dirac_polarization_wave_checks.py constructs the full containing Clifford fields, checks (4) at spatial points for arbitrary complex amplitudes, and integrates the canonical norm over the bound coordinates. Independent source integrals and finite spatial derivatives verify the linear polarization modes.

For the finite two-wave solutions it recomputes the scalar moment and checks the complete nonlinear equation using independent finite derivatives in time, the normal coordinate and every bound coordinate. It also checks both normalization factors, the normal current and the changing bound current. Conditions remain active under Python optimization.

Related

• Normal propagation of spatial Dirac modes — supplies the full normal partner and boost convention.
• Scalar envelope of a bound Dirac field — supplies the profile response when total scalar density changes.
• Rotation-neutral Dirac fields — supplies the complete polarized bound field and its currents.
• Polarized Dirac shape response — supplies additional spatial shape motions beyond this radial polarization family.
• Self-consistent scalar Dirac moment modes — supplies the source equation and separate radial stability limits.