Dirac source linearization

Summary: The written scalar Dirac self-interaction gives an explicit real-linear response with ordinary and conjugate mode couplings determined by the higher spatial field.

The existing nonlinear term

Use the scalar self-interaction present in the projected Dirac action. In units $\hbar=1$, its Hamiltonian form is

$$i\partial_t\Psi=H_0\Psi+\mathcal N[\Psi], \qquad \mathcal N[\Psi]=\kappa(\Psi^\dagger\beta\Psi)\beta\Psi. \tag{1}$$

Here $\beta=\beta^\dagger$, $\beta^2=I$, $\kappa$ is real, and $H_0$ is the specified self-adjoint spatial Dirac operator including whichever fixed background terms have actually been adopted. Its domain and the spatial measure must make the following pairings valid. The energy is

$$\mathcal E[\Psi] =\langle\Psi,H_0\Psi\rangle +\frac\kappa2\int(\Psi^\dagger\beta\Psi)^2d\mu. \tag{2}$$

This is one field on the nested macroscopic spatial manifold. Spinor components describe amplitudes over that space, not independent spatial dimensions. Equation (1) is a specified contribution to V2 dynamics, not the completed master action or a derivation of occupied particle geometries. No mediator particle is introduced.

Nonlinear Dirac equations with this scalar structure and linearization about solitary waves are treated in Boussaïd and Comech. The sign convention here is fixed directly by (2).

Differentiate the complete source

At a spatial field $\phi$, set

$$s=\phi^\dagger\beta\phi\in\mathbb R, \qquad v=\beta\phi.$$

For a variation $\eta$, the scalar changes by $\delta s=v^\dagger\eta+\eta^\dagger v$. Therefore

$$D\mathcal N[\phi]\eta =\kappa\left[s\beta\eta +v(v^\dagger\eta+\eta^\dagger v)\right]. \tag{3}$$

Equivalently,

$$\boxed{ D\mathcal N[\phi]\eta=A_N\eta+B_N\overline\eta, \qquad A_N=\kappa(s\beta+vv^\dagger), \qquad B_N=\kappa vv^T.} \tag{4}$$

The $s\beta$ term is the variation of the final field factor. The two rank-one terms come from changing the source scalar itself. Retaining only $s\beta\eta$ would freeze the higher particle's source while it is being perturbed. The complete derivative is real-linear; generally $D\mathcal N[i\eta]\ne iD\mathcal N[\eta]$.

The matrices obey $A_N=A_N^\dagger$ and $B_N=B_N^T$. Spatial variation of $s(x)$ and $v(x)$ retains the target's local shape and polarization. Replacing them by spatial averages is an additional approximation.

Linearized dynamics about a stationary profile

Suppose $\Psi_0=e^{-i\mu t}\phi$ is an actual solution of (1), so

$$(H_0+\kappa s\beta)\phi=\mu\phi. \tag{5}$$

In the same rotating frame put $\Psi=e^{-i\mu t}(\phi+\eta)$. The linearized equation is

$$i\partial_t\eta=A\eta+B\overline\eta, \qquad A=H_0-\mu+A_N, \qquad B=B_N. \tag{6}$$

Its doubled representation is

$$i\partial_t \begin{pmatrix}\eta\\\overline\eta\end{pmatrix} =\mathcal L \begin{pmatrix}\eta\\\overline\eta\end{pmatrix}, \qquad \mathcal L= \begin{pmatrix}A&B\\-\overline B&-\overline A\end{pmatrix}. \tag{7}$$

The second row is the conjugate of the first; it does not add another physical field or another set of spatial dimensions. With $\Sigma_3=\operatorname{diag}(I,-I)$,

$$\mathcal H=\Sigma_3\mathcal L =\begin{pmatrix}A&B\\\overline B&\overline A\end{pmatrix} =\mathcal H^\dagger. \tag{8}$$

This is the quadratic-energy structure. $\mathcal L$ itself need not be Hermitian or have a real spectrum. Stability cannot be inferred merely from the Hermiticity of $H_0$. The global phase tangent supplies the consistency identity

$$\mathcal L \begin{pmatrix}i\phi\\-i\overline\phi\end{pmatrix}=0, \tag{9}$$

which follows immediately from (5). A moving orbit or a background with several temporal frequencies instead requires time-dependent coefficients; a stationary spectral analysis cannot silently replace that motion.

Spatial coupling between mode families

Let $f_a(x)$ be photon-sector test modes and $u_\lambda(x)$ target variation modes, normalized by the action's kinetic metric. The ordinary and conjugate matrix elements of the interaction derivative are

$$K_{a\lambda}=\int f_a^\dagger A_Nu_\lambda\,d\mu, \qquad C_{a\lambda}=\int f_a^\dagger B_N\overline{u_\lambda}\,d\mu. \tag{10}$$

Thus a target amplitude vector $q$ produces the linearized source $Kq+C\overline q$. The spatial integrals include polarization, relative phase, and all of the higher-dimensional profile present in (4). Any cross-family kinetic matrix element of $H_0$ must also be retained.

Interchanging target and photon mode labels gives Hermitian reciprocity for $K$ and transpose symmetry for $C$:

$$K_{\lambda a}=\overline{K_{a\lambda}}, \qquad C_{\lambda a}=C_{a\lambda}. \tag{11}$$

For the second equality, use $B_N^T=B_N$ and transpose the scalar pairing. In the doubled mode space these are precisely the symmetries of the real quadratic energy. They do not require treating the conjugate variation as an independent particle.

A zero of one matrix element excludes that particular linear response. It is not a zero of the complete interaction unless all relevant ordinary, conjugate, and kinetic blocks vanish. The criterion must be applied to the actual nonlinear source, not to the parity of a bare field alone; angular kernel parity gives an explicit example of this distinction.

Which temporal frequencies the source contains

For a finite prescribed field $h(x,t)=\sum_j c_j\phi_j(x)e^{-i\Omega_jt}$, equation (1) gives

$$\mathcal N[h] =\kappa\sum_{j,\ell,m} \overline c_jc_\ell c_m (\phi_j^\dagger\beta\phi_\ell)\beta\phi_m e^{-i(\Omega_\ell+\Omega_m-\Omega_j)t}. \tag{12}$$

Projection onto an output mode $f_a$ uses the full cubic spatial overlap

$$T_{a;j\ell m} =\kappa\int f_a^\dagger\beta\phi_m (\phi_j^\dagger\beta\phi_\ell)\,d\mu. \tag{13}$$

The frequency combination and spatial compatibility follow from the same written term. Frequency matching alone does not make (13) nonzero. Several triples reaching the same output frequency interfere coherently.

Equations (10)–(13) assume a linear output-mode inclusion in the master field. If a photon is instead realized as a nonlinear collective observable of that field, derive its tangent map and induced equations. For example a bilinear observable contains frequency differences, whereas the cubic field source in (12) contains the displayed three-term combinations. Their different frequency rules cannot be interchanged without deriving that output map.

Scope of the concrete calculation

Complete scalar response determines the full pointwise rank, singular values and inverse of (3). At nonzero scalar density the complete source derivative is invertible; at scalar nullity it retains one real response direction. Those facts concern the full source before a spatial channel projection and before adding $H_0-\mu$.

Transition-density coupling derives the corresponding nonlocal source derivative and full quartic mode tensor for the projected bilocal density interaction, with both patterns retained.

The operator $D\mathcal N[\phi]$ is now explicit for this term. The remaining physical inputs are the actual higher-sector solution, the photon embedding or collective map, all additional action terms, and the boundary conditions selecting incoming and outgoing modes. The matrix formulas apply at any depth supporting the specified Dirac structure and do not themselves single out depth five.

Local scalar contact normalization adds the spatial virial and an exact normalized massive profile for the same local source convention. It makes explicit why a scalar vertex coefficient does not by itself determine a stationary particle frequency.

Verification

Local polarized pulse coupling retains the exact cross-current of two traveling components of this same field. Its complete local equations give source zeros and both reciprocal weak-scattering maps without first postulating a bilocal spatial kernel.

Causal scalar response solves this complete derivative about a specified uniform spatial background, including every scalar angular mode, both signed propagation channels and the conserved zero mode. The background is a control, not an identified particle.

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Source frequencies in spatial coupling tracks the distinct temporal frequencies in the bilinear density, complete cubic field source and perturbations relative to the stationary carrier.

Related

• Descending photon source — retains normal propagation when projecting the higher-field source.
• Projected Dirac current — supplies the written scalar interaction and its action conventions.
• Canonical coupling normalization — fixes the mode metrics used in (10).
• Angular kernel parity — distinguishes source selection from parity of the underlying field.