Canonical spatial spin response

Summary: The full moving spatial frame fixes the canonical particle center and relates its electric spin-orbit response to the magnetic moment without an independent coefficient.

Retain the entire occupied polarization

The conditional spatial current calculation supplies a normalized radial Dirac profile on $d$ ordinary spatial coordinates. Its centered scalar binding determines the translation inertia $\mathcal I$ and current moment $F_0$. With a normalized complex vector $z$ in the complete occupied multiplet, write

$$J_{ij}=z^\dagger B_{ij}z,\qquad B_{ij}=W_{\rm rest}^\dagger(-i\alpha_i\alpha_j)W_{\rm rest}, \qquad g_{\rm mag}=4\mathcal I F_0. \tag{1}$$

Here $W_{\rm rest}$ embeds the upper radial polarization at rest. The full spatial field includes its lower radial component. These matrices describe spatial rotations and add no physical dimensions. The complete polarization description keeps the whole vector $z$ even when its bivector $J$ does not determine every higher-dimensional polarization observable.

The soft current derivation gives the velocity tangent $\zeta_i=iMx_i\phi+\alpha_i\phi/2$ and the number centroid $R_\rho=X+F_0Jv$. The charge $q$ and minimal connection of that same field remain conditional inputs. Work to first order in a weak electric field and to the leading nontrivial order in the particle velocity.

Normalize the moving spatial frame

Let $W(x)$ contain every normalized rest profile as columns, and let $Z_i(x)=iMx_iW+\alpha_iW/2$. Full spatial integration gives matrix identities

$$\int W^\dagger W=I,\qquad \int W^\dagger Z_i=0,\qquad K_{ij}-K_{ji}=-\frac i2B_{ij},\qquad K_{ij}=\int Z_i^\dagger Z_j. \tag{2}$$

The first mixed integral vanishes by parity. The last identity is the matrix version of the velocity symplectic curvature; it holds before choosing the occupied ray.

Normalize $W_v=W+v_iZ_i$ by its integrated Gram matrix: $\widetilde W_v=W_v(\int W_v^\dagger W_v)^{-1/2}$. For the field $\widetilde W_vz$, the term containing $dz$ remains the canonical $\tfrac i2(z^\dagger dz-dz^\dagger z)$. Equation (2) fixes the leading velocity connection as $i\int\widetilde W_v^\dagger d\widetilde W_v =\tfrac14v_iB_{ij}dv_j+\cdots$. A local frame gauge can absorb the symmetric connection terms of a higher-order profile completion; the curvature at rest is fixed by its actual first tangents.

Together with the translation momentum, the retained action one-form is

$$\theta=\frac i2(z^\dagger dz-dz^\dagger z) +\mathcal I v\cdot dX+\frac14v_iJ_{ij}dv_j. \tag{3}$$

This states the orders needed below, not an all-velocity canonical description of the scalar-bound particle.

Find the canonical spatial center

Define

$$\boxed{ P=\mathcal I v,\qquad r=X+\frac{Jv}{4\mathcal I}.} \tag{4}$$

Then $P\cdot dr=\mathcal I v\cdot dX+\tfrac14v\cdot J\,dv$. This remains true when $z$ changes: the extra term is $\tfrac14v_i(dJ_{ij})v_j=0$ because $dJ$ is antisymmetric. Thus (3) becomes the usual $P\cdot dr$ plus the full polarization one-form. No frozen-spin assumption is needed for this coordinate identity.

The physical number centroid is consequently

$$\boxed{ R_\rho=r+\frac{g_{\rm mag}-1}{4\mathcal I^2}JP.} \tag{5}$$

The scalar center, canonical center and number centroid describe different moments of the same moving spatial field. They are not separate physical spaces or a division of the six-dimensional electron.

Obtain the electric spin-orbit coefficient

For a uniform electric field, the actual charge energy is $-qE\cdot R_\rho$. Its velocity-dependent part in canonical coordinates is

$$\boxed{ H_{\rm SO} =-\frac{q(g_{\rm mag}-1)}{4\mathcal I^2} \sum_{i<j}J_{ij}(E_iP_j-E_jP_i).} \tag{6}$$

For a chosen three-dimensional measurement plane, $J_{ij}=2\epsilon_{ijk}S_k$, so

$$H_{\rm SO} =-\frac{q(g_{\rm mag}-1)}{2\mathcal I^2} \,S\cdot(E\times P). \tag{7}$$

The same current gives the rest magnetic term $H_B=-qg_{\rm mag}S\cdot B/(2\mathcal I)$ in that plane. With the conventional coefficient names $c_F=g_{\rm mag}/2$ and $c_S=g_{\rm mag}-1$,

$$\boxed{c_S=2c_F-1.} \tag{8}$$

This agrees with the low-velocity spin-half electromagnetic coefficient relation derived from Lorentz invariance; see the explicit Lagrangian and matching discussion in Paz, An Introduction to NRQED, equation (3). The comparison concerns the response coefficients of an individual particle. It imports no mediator-particle interpretation and does not model an atom.

There is a direct consistency check. Put $c=(g_{\rm mag}-1)/(4\mathcal I^2)$ and keep $J$ fixed at linear order in the applied field. Equations (5)–(6) give $\dot P=qE$, $\dot r=P/\mathcal I+qcJE$ and therefore $\dot R_\rho=v+2qcJE$. This is exactly the full spatial current derived independently from the complete source resolvent.

The result fixes the leading uniform-field spin-orbit response. Spatial field gradients, the Darwin term, finite-frequency deformation and an all-velocity spin-precession law require further calculation. The scalar six-dimensional ground profile still gives $g_{\rm mag}\simeq2.07691916$, so this coefficient relation does not repair its mismatch with the electron magnetic moment. The same parameters also control the soft forward scattering term; these related tests must be satisfied together.

Verification

soft_photon_response_checks.py integrates (2) as full spatial matrices at depths three and six, normalizes the moving frame, and independently differentiates its overlap to recover the Berry connection. Halving the velocity tests the omitted order. Arbitrary complex polarization variations check (4), while the canonical equations are compared with the independently derived spatial current. All checks remain active under Python optimization.