Canonical coupling normalization

Summary: Kinetic Gram matrices fix normalized transition vertices and conserved generators, separating charge universality from profile-dependent interaction strength.

Modes and the kinetic metric

Use finite sets of linearly independent, admissible mode sections in arbitrary nested physical sectors. Their coefficient labels are not additional spatial dimensions. Let $S_d$ map coefficients $c_d$ to the spatial field, and let the positive time-kinetic form induce

$$Z_d=S_d^\dagger S_d>0,\qquad L_{\rm kin}=\sum_d i c_d^\dagger Z_d\dot c_d.$$

The dagger includes the specified spatial measure and bundle inner product. The block form assumes kinetic orthogonality between the retained families. Spatial nesting alone does not imply it. If cross terms occur, use the full positive Gram matrix for the independent combined basis and normalize that matrix. Redundant sections must first be removed; a singular Gram matrix cannot be normalized with an inverse.

With $\hbar=1$, define $z_d=Z_d^{1/2}c_d$. The kinetic term becomes $i z_d^\dagger\dot z_d$. For a coherent cross-sector vertex with a single real perturbation amplitude $A$ of the same field,

$$U_{\rm tr}=A\,c_a^\dagger K_{ab}c_b+\text{complex conjugate},$$

Here $A$ is a retained variation of $\Psi_\infty$, and the trilinear term is conditional on its expansion from that field's action. It does not introduce a separate interaction field or a mediator particle. The matter-normalized vertex is

$$\widehat K_{ab}=Z_a^{-1/2}K_{ab}Z_b^{-1/2}. \tag{1}$$

If $A$ has time-kinetic coefficient $Z_A>0$, its canonical amplitude is $A_c=\sqrt{Z_A}A$, and the fully normalized vertex is $\widehat K_{ab}/\sqrt{Z_A}$. Matrix-valued response metrics require the corresponding transformation on their own index.

For a density source $j_d=c_d^\dagger T_dc_d$, the transformation instead is

$$j_d=z_d^\dagger\widehat T_dz_d,\qquad \widehat T_d=Z_d^{-1/2}T_dZ_d^{-1/2}. \tag{2}$$

Each current has two matter legs. A current-current interaction therefore cannot be normalized by applying (1) just once to its matrix of scalar coefficients. Measures and spatial maps must be transformed consistently.

Independence of coefficient conventions

Under any invertible basis change $c_d=T_d'\widetilde c_d$,

$$\widetilde Z_d=T_d'^\dagger Z_dT_d',\qquad \widetilde K_{ab}=T_a'^\dagger K_{ab}T_b'.$$

The matrix

$$U_d=Z_d^{1/2}T_d'\widetilde Z_d^{-1/2}$$

is unitary, because $U_d^\dagger U_d=I$. Substitution proves

$$\widetilde{\widehat K}_{ab}=U_a^\dagger\widehat K_{ab}U_b. \tag{3}$$

Thus rank and singular values of the normalized vertex are independent of the arbitrary normalization of its basis sections. Its entries still depend on which physical states are used; mass eigenbases and their relative phases have to be determined to predict particular transitions. The existing CKM calculation is a concrete application.

For a single mode per family, $|K_{ab}|/\sqrt{Z_aZ_bZ_A}$ is invariant. The pair energy $g_{ab}j_aj_b$, a transition vertex, and a mass-operator coefficient remain different objects even if all are called couplings.

Generators, charge, and selection

Let a self-adjoint generator $\mathsf Q$ preserve each retained mode subspace, so

$$\mathsf Q S_d=S_d q_d,\qquad q_d^\dagger Z_d=Z_dq_d.$$

The normalized generator is Hermitian:

$$\widehat q_d=Z_d^{1/2}q_dZ_d^{-1/2}.$$

If the underlying generator is the same scalar $q$ on two families, $\mathsf Q S_a=qS_a$ and $\mathsf Q S_b=qS_b$, then their unnormalized charge matrices are $qZ_a,qZ_b$. After normalization both become $qI$, regardless of spatial depth, profile width, or the number of retained modes. This proves preservation of a common charge under projection and normalization. It does not derive which master generator is electric charge, its eigenvalues, or why both physical sectors carry the same one.

A neutral field perturbation preserves this charge when

$$\widehat q_a\widehat K_{ab}=\widehat K_{ab}\widehat q_b. \tag{4}$$

For charge eigenvectors with eigenvalues $q_i,q_j$, (4) reads $(q_i-q_j)\widehat K_{ij}=0$. A charged complex field amplitude transforming as $A\mapsto e^{i q_A\theta}A$, with matter transforming as $z_d\mapsto e^{i\widehat q_d\theta}z_d$, changes the criterion to

$$\widehat q_a\widehat K_{ab}-\widehat K_{ab}\widehat q_b =q_A\widehat K_{ab}. \tag{5}$$

This is the infinitesimal invariance of $A z_a^\dagger\widehat K_{ab}z_b$. It is a selection rule, not a numerical prediction of an allowed entry. For non-Abelian generators the perturbation index transforms too, giving an invariant tensor rather than a single charge-difference equation.

Universal charge and resolved spatial structure

For a normalized positive density $\rho_d$ carrying the same charge $q$, the projected spatial form factor is

$$F_d(k)=q\int_{Q_d}e^{ik\cdot r(x)}\rho_d(x)\,d\mu_d(x), \qquad F_d(0)=q.$$

Here $r$ is the specified readout onto shared spatial coordinates. The zero-momentum identity uses the complete conserved marginal. It does not require all the $F_d(k)$ to agree at nonzero momentum. For a centered Gaussian shared marginal with covariance $C_d$,

$$F_d(k)=q\,e^{-k^{\mathsf T}C_dk/2}.$$

Thus equal charge is consistent with unequal spatial response and contact integrals. A mode count or spatial width need not multiply a normalized particle's charge. This statement concerns an assumed conserved charge; neutral composite distributions or other electromagnetic operators need their own form factors.

The quantum-field benchmark is charge universality in the Thomson limit, described in PDG, Electroweak Model, §10.3.1. The calculation above is an action-level projection identity, not an all-orders proof of that benchmark for V2. A common gauge-field kinetic normalization and a gauge-consistent elimination of omitted modes remain necessary. The projected current note gives the exact local conservation defect when a truncation drops such modes.

Related and verification

• Generalized sector coupling — the theory-wide calculation and its physical inputs.
• Nested sector reciprocity — fixes the spatial pairing used in the raw vertex.
• Common field response — applies response factorization only after the source convention is specified.
• Neutrino channel control — applies charge selection without suppressing every interaction.

generalized_coupling_checks.py tests nonorthogonal bases, nonunitary basis changes, conserved generators, charged transition selection, and independently integrated form factors.