Soft spatial current response

Summary: The full spatial translation and velocity tangents determine the low-frequency electric current, its polarization correction, and the magnetic moment of a conditionally gauged scalar Dirac profile.

Specify the physical current being tested

Let a normalized localized field on $d$ ordinary spatial coordinates obey

$$H_b\phi=M\phi,\qquad H_b=\alpha_i(-i\partial_i)+\beta(m+a|x|^2),\qquad \int\phi^\dagger\phi\,dx=1. \tag{1}$$

Its quadratic scalar source is varied with its own centered moment functional. The coordinates belong to the one nested macroscopic spatial manifold. The field components describe its spatial polarization.

For this calculation, make an explicit additional hypothesis: a retained connection amplitude of the same field enters the equation as

$$H[A,\Phi]=\alpha_i(-i\partial_i-qA_i) +\beta\mathfrak m[\Psi]+q\Phi. \tag{2}$$

The connection, charge generator $qI$, and its eventual radiation normalization must be derived from the master action to identify this with a physical interaction. Equation (2) is a conditional test of the resulting particle response. It introduces no mediator particle.

Varying this same action gives the reciprocal connection source

$$\rho_{\rm em}=q\Psi^\dagger\Psi,\qquad j_{{\rm em},i}=q\Psi^\dagger\alpha_i\Psi. \tag{3}$$

The scalar binding functional is invariant under local common phase changes, so it preserves the corresponding number continuity equation. Keeping the full spatial field also keeps the kinetic normalization required by Dressed source vertices.

Keep the field's displacement and velocity

Write the normalized radial field as

$$\phi(x)=\frac{f(r)w+i g(r)\alpha(\widehat x)w} {\sqrt{\Omega_{d-1}}},\qquad \beta w=w,\quad w^\dagger w=1. \tag{4}$$

The constant vector $w$ may have arbitrary complex polarization in the selected complete spatial multiplet. Define

$$J_{ij}=w^\dagger(-i\alpha_i\alpha_j)w,\qquad G=\int_0^\infty g(r)^2r^{d-1}dr, \qquad F_0=\frac2d\int_0^\infty r^d f(r)g(r)\,dr. \tag{5}$$

The radial Dirac equations imply

$$F_0=\frac{1-2G/d}{2M}. \tag{6}$$

This current-moment identity is derived in Normal propagation of spatial Dirac modes. The total spatial angular momentum includes the orbital contribution and satisfies $\langle\mathcal L_{ij}\rangle=J_{ij}/2$ on (4).

The complete translation calculation supplies the two tangent fields

$$\eta_i=-\partial_i\phi,\qquad \zeta_i=iMx_i\phi+\frac12\alpha_i\phi, \qquad \Psi=e^{-iMt}[\phi+X_i\eta_i+v_i\zeta_i+\cdots]. \tag{7}$$

Here $X$ is the scalar center and $v$ its generalized velocity coordinate. The first variation of every scalar binding source under $v_i\zeta_i$ vanishes pointwise. The profile's inertial coefficient is

$$\mathcal I=M+\frac{T}{d},\qquad T=\langle\phi,\alpha_i(-i\partial_i)\phi\rangle. \tag{8}$$

The stationary frequency $M$ and translation inertia $\mathcal I$ must not be interchanged. For the zero-bare-mass quadratic scalar branch, $\mathcal I=M(1+2/(3d))$.

Derive the spatial symplectic coefficients

The original field action has kinetic term $\tfrac i2\int(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi)\,dx$. Direct spatial integration gives

$$\boxed{ \begin{aligned} 2\operatorname{Im}\langle\eta_i,\eta_j\rangle&=0,\\ 2\operatorname{Im}\langle\eta_i,\zeta_j\rangle &=\mathcal I\delta_{ij},\\ 2\operatorname{Im}\langle\zeta_i,\zeta_j\rangle &=-\frac12J_{ij}. \end{aligned}} \tag{9}$$

The first identity follows from commuting translation generators. The second includes the full Dirac velocity partner in (7). For the third, put $\chi_i=Mx_i\phi-i\alpha_i\phi/2$, so $\zeta_i=i\chi_i$. Its cross terms give $-MF_0J_{ij}$ to the imaginary overlap, while the Dirac spin term gives

$$\frac14\langle\phi,-i\alpha_i\alpha_j\phi\rangle =\frac14\left(1-\frac{4G}{d}\right)J_{ij}. \tag{10}$$

Equation (6) makes their sum $-J_{ij}/4$, proving the last line of (9). The orbital and lower radial components are essential to these identities.

The number-density centroid differs from the scalar center. Since $2\operatorname{Re}(\phi^\dagger\zeta_j)=\phi^\dagger\alpha_j\phi$,

$$\boxed{R_{\rho,i}=X_i+F_0J_{ij}v_j.} \tag{11}$$

Indeed, $\int x_i\phi^\dagger\alpha_j\phi\,dx=F_0J_{ij}$. Number continuity then identifies the integrated number current with $\dot R_\rho$. All quantities in (9)–(11) are integrals of fields and derivatives on the actual spatial coordinates.

Apply a slowly varying uniform electric field

To the required order, use energy $\mathcal I|v|^2/2-qE\cdot R_\rho$. The symplectic matrix from the stated field-action convention is

$$\begin{pmatrix} 0&-\mathcal I I_d\\ \mathcal I I_d&J/2 \end{pmatrix} \tag{12}$$

in coordinates $(X,v)$. Its equations give

$$\mathcal I\dot v=qE,\qquad \mathcal I\dot X+\frac12J\dot v =\mathcal I v+qF_0JE. \tag{13}$$

Keeping $J$ at its unperturbed value suffices at linear order in this uniform electric perturbation. Therefore

$$\dot R_\rho =v+q\left(\frac{2F_0}{\mathcal I} -\frac1{2\mathcal I^2}\right)JE. \tag{14}$$

For time dependence $e^{-i\omega t}$ and $E=i\omega A$, the induced current is

$$\boxed{ \delta j_{\rm em} =-q^2\left[ \frac1{\mathcal I}I_d +i\omega\left(\frac1{2\mathcal I^2} -\frac{2F_0}{\mathcal I}\right)J +O(\omega^2)\right]A.} \tag{15}$$

The leading coefficient is independent of target polarization. It gives the usual Thomson current structure if $\mathcal I$ is the physical inertial mass and (2) is the physical normalized connection coupling. Those identifications are additional to the derivation.

Check against the complete source response

There is an independent route which keeps the entire radial spectrum. Let $c_i=\beta x_i\phi$ and $u_i=\alpha_i\phi$ be the scalar-center and current source columns. For any Hermitian source labels $A,B$, define

$$\Pi_{AB}(\omega)= \langle A\phi,(\omega-H_b+M)^{-1}B\phi\rangle +\langle B\phi,(-\omega-H_b+M)^{-1}A\phi\rangle. \tag{16}$$

Both signed frequency terms are required. The center curvature is $\kappa=-2a/Q_\sigma$, where $Q_\sigma=\int\phi^\dagger\beta\phi\,dx$. Its feedback gives

$$\boxed{\chi_{uu}=\Pi_{uu} +\Pi_{uc}\kappa(I-\Pi_{cc}\kappa)^{-1}\Pi_{cu}.} \tag{17}$$

All coordinate indices are retained. Parity separates these odd source columns from the even scalar shape sources at zero spatial wavenumber. The current columns can also be generated without a current ansatz:

$$u_i=i(H_b-M)(x_i\phi). \tag{18}$$

Complete angular radial families in (16)–(18) reproduce every coefficient of (15). The frozen-profile response $\Pi_{uu}$ tends to zero; the moving center supplies the finite leading term. This distinguishes a vanishing frozen overlap from a vanishing physical response.

Relate the current moment to a measured particle property

The actual antisymmetric magnetic moment tensor from (3) is

$$\mu_{ij}=\frac12\int(x_i j_{{\rm em},j} -x_j j_{{\rm em},i})\,dx=qF_0J_{ij}. \tag{19}$$

Using $S_{ij}=J_{ij}/2$ and defining the gyromagnetic factor by $\mu_{ij}=g_{\rm mag}qS_{ij}/(2\mathcal I)$ gives

$$\boxed{g_{\rm mag}=4\mathcal I F_0,\qquad \chi_{uu}(\omega,0) =\frac1{\mathcal I}I_d +i\omega\frac{1-g_{\rm mag}}{2\mathcal I^2}J+O(\omega^2).} \tag{20}$$

For a selected three-dimensional observation plane this is the ordinary magnetic-moment convention. Selecting that plane distinguishes a measurement geometry; it introduces no inherent split in an electron's six spatial directions.

The zero-bare-mass scalar ground profiles give:

A single-electron measurement in 2023 found $g/2=1.00115965218059(13)$, hence $g=2.00231930436118(26)$. This is the benchmark reported by Fan, Myers, Sukra and Gabrielse, not an atomic-system calculation.

Thus the stated six-dimensional scalar profile, with the current (3), does not reproduce the electron magnetic moment. Changing the common coupling rescales the zero-bare-mass profile and leaves this dimensionless factor unchanged. A different occupied field or a derived modification of the physical current would have to supply its own prediction.

The spatially uniform kernel (20) is not yet the on-shell photon amplitude. The photon phase and the complete degenerate target polarizations supply another term, calculated in Soft polarized scattering.

Conserved spatial charge gives the full kinetic conditions a proposed geometric charge generator must satisfy.

Canonical spatial spin response uses the full moving polarization frame to derive the canonical center and the electric spin-orbit coefficient from this same current moment.

Verification

soft_photon_response_checks.py checks (9)–(11) by direct radial and angular integration of the full spatial fields, including arbitrary complex polarizations. A separate angular radial resolvent calculation verifies (15) and (20) at depths 3, 5, 6 and 10, with zero and positive bare scalar mass. The companion scattering check retains the photon phase and all scalar shape feedback. Reports have timestamped names and checks remain active under Python optimization. These checks establish the stated conditional mathematics; the connection, charge assignment, stable occupied particles and empirical interaction law remain to be derived.