Translational Dirac response

Summary: Varying the scalar center restores the exact translation zero mode and yields a calculable inertial response of the full self-bound spatial Dirac field.

A self-bound field must be able to move

The centered scalar interaction determines its own spatial profile and center. Holding that center fixed while varying the field would replace part of its dynamics by an external constraint. This note retains the center and derives its response from the same energy as the scalar Dirac moment modes.

All coordinates are ordinary spatial coordinates in a sector of the one nested physical manifold. Translation moves the field within those coordinates. Response coefficients and angular functions add no physical dimensions or interaction particles.

Use $\sigma=\Phi^\dagger\beta\Phi$, $Q_\sigma=\int\sigma\ne0$, $P_\sigma=\int x\sigma$, $X=P_\sigma/Q_\sigma$, and

$$M_\sigma=\int(x-X)(x-X)^T\sigma\,dx, \qquad \mathcal E_{\rm int}=\frac{g_0}{2}\operatorname{tr}M_\sigma^2. \tag{1}$$

$Q_\sigma$ is a scalar bilinear integral, not an electric charge. It need not equal the separately normalized field norm.

The center contributes a required dipole feedback

At a centered radial background, $M_\sigma=(T/d)I$ and $a=g_0T/d$. For an arbitrary field variation let

$$\delta\sigma=2\operatorname{Re}(\Phi^\dagger\beta\eta),\qquad \delta P=\int x\,\delta\sigma\,dx,\qquad \delta C=\int xx^T\delta\sigma\,dx.$$

The first variation of the central moment is $\delta M=\delta C$, but the source also depends on $x-X$. Differentiating the entire scalar potential gives

$$\boxed{\delta U(x)=g_0x^T\delta Cx -\frac{2a}{Q_\sigma}x\cdot\delta P.} \tag{2}$$

The second term supplies the center's dipole response. Its coefficient $\kappa=-2a/Q_\sigma$ follows from the same centered energy, including its sign; it is not an independently chosen interaction strength.

For two variations the full interaction Hessian is

$$\begin{aligned} \mathcal E_{\rm int}''(\eta,\zeta) ={}&2\operatorname{Re}\langle\eta,\beta ar^2\zeta\rangle +g_0\operatorname{tr}(\delta C_\eta\delta C_\zeta)\\ &-\frac{2a}{Q_\sigma}\delta P_\eta\cdot\delta P_\zeta. \end{aligned} \tag{3}$$

This is symmetric in both variations, as required for reciprocal dynamics. The negative center term is consistent with translation invariance: a shift of the entire field does not cost binding energy.

The dipole source has a full spatial Dirac decomposition

With the rest polarization $w$ and Clifford matrices of the profile note, a dipole source in direction $b$ is $v_b=\beta(b\cdot x)\Phi$. Its angular polynomial splits as

$$\begin{aligned} p_b(x)&=(b\cdot x)w=M_b(x)+\alpha(x)N_b,\\ N_b&=\frac1d\alpha(b)w,\qquad M_b(x)=(b\cdot x)w-\frac1d\alpha(x)\alpha(b)w. \end{aligned} \tag{4}$$

Both $M_b$ and $N_b$ are annihilated by $\alpha\cdot\partial$. The radial operators and source norms follow by the same direct method as Polarized Dirac shape response, now with degrees one and zero. For the circulation tensor $J_{ij}=w^\dagger(-i\alpha_i\alpha_j)w$ their angular Gram matrices are

$$G_N=\frac{I+iJ}{d^2},\qquad G_M=\frac Id-G_N. \tag{5}$$

The two radial operators are $H_{d+2}$ and $-XH_dX$, with the same physical scalar coefficients $m,a$. The larger numerical radial parameter again accounts only for powers of the existing radius.

Let $L$ contain these two operators minus the stationary frequency. For helicity $\lambda$ of $iJ$, let $u_\lambda,u_{-\lambda}$ be the normalized radial source vectors with squared angular weights $1/d-(1+\lambda)/d^2$ and $(1+\lambda)/d^2$. Their bare response is

$$\chi_\lambda(z)=u_\lambda^T(z-L)^{-1}u_\lambda +u_{-\lambda}^T(-z-L)^{-1}u_{-\lambda}. \tag{6}$$

The complete dipole response is $\chi_\lambda/(1-\kappa\chi_\lambda)$, with the center coefficient from (2). The ordinary and conjugate blocks are both retained.

Translation gives an exact response identity

For a translated stationary field $\Phi(x-\epsilon b)$, the tangent is $\eta_b=-b\cdot\nabla\Phi$. Translation leaves the central moment unchanged and gives $\delta P=Q_\sigma b$. Differentiating the stationary equation therefore gives

$$\boxed{\kappa\chi_\lambda(0)=1.} \tag{7}$$

This places a zero-frequency collective mode in the complete response, even though the frozen confining Dirac operator has finite excitation gaps. The identity holds in all physical translation directions.

Commuting translations also have zero mutual symplectic pairing. In the radial representation this is

$$\|L^{-1}u_\lambda\|^2=\|L^{-1}u_{-\lambda}\|^2, \qquad \chi_\lambda'(0)=0. \tag{8}$$

The first identity follows equivalently from the spatial integral of the commutator of momentum generators. It is a useful independent check of the angular decomposition and both Dirac radial branches.

At finite numerical resolution the translation zero and its conjugate momentum direction can produce small spurious growth rates. Fixing total momentum and quotienting the translation direction removes that pair. This operation preserves the remaining fluctuation frequencies; it does not remove a nonzero physical instability.

The inertial coefficient follows from the low-frequency response

Apply a specified slowly varying scalar source $h(t)\cdot x\,\beta$. Its contribution to the energy is $h\cdot P_\sigma$ and the force on the translated field is $-Q_\sigma h$ to first order. Expanding (6) at zero frequency and using (7)–(8) gives

$$\chi^{\rm full}_\lambda(z) =-\frac{2}{\kappa^2\chi_\lambda''(0)z^2}+O(z^{-1}). \tag{9}$$

Comparing $\delta X=\delta P/Q_\sigma$ with the force response defines the inertial coefficient

$$\boxed{\mathcal M_{\mathrm{tr}} =-\frac{\kappa^2Q_\sigma^2}{2}\chi_\lambda''(0) =-2a^2\chi_\lambda''(0).} \tag{10}$$

The second derivative involves the sum of opposite helicities, so this leading inertia is isotropic for the radial background. Higher-frequency response can still depend on $J$. In the present examples it is positive.

At $a=1$, $m=0$, the numerical values are approximately $2.080669220$ at depth three, $2.770370932$ at five, $3.084232390$ at six, and $4.210308101$ at ten. These are coefficients in the specified model's translation dynamics. The instantaneous nonlocal energy does not supply a Lorentz-covariant completion, so equality to the stationary frequency or total binding energy is not presumed. At depths with a separate radial instability, this translation response does not make the whole background stable.

Relation to a force between two patterns

This calculation supplies a field's response when a reciprocal pair interaction exerts a force. The force itself must come from differentiating that pair energy with respect to its two positions. A moment pairing which depends only on separately centered shapes and a fixed identification matrix has no dependence on their relative translation. Its orientation and shape responses remain meaningful, while a separation-dependent force requires the corresponding spatial contact or propagation kernel.

Verification

angular_dirac_response_checks.py checks the dipole source by full Clifford angular integration and the physical radial equations. It verifies the translation identity and the vanishing first frequency derivative, then compares the remaining spectra at increased radial resolution. Independent finite differences of the centered full-field energy and source check (2)–(3), including the sign of the center contribution.

Related

• Centroid-relative self-binding — derives the center variation in the scalar spatial model.
• Self-consistent scalar Dirac moment modes — supplies the full first-order stationary profile.
• Polarized Dirac shape response — derives the angular and ordinary/conjugate response method used here.