Local scalar contact on physical field derivatives

Summary: Prolonging the written local scalar interaction to physical field derivatives repeats its local source blocks without generating a simplex coupling, and finite derivative data do not close the spatial kinetic equation.

Test the local interaction before changing its action

The mass-operator construction identifies the filtered derivative symbols of the physical field with a cumulative symmetric carrier. That identification gives the correct Hockey-Stick rank. It does not yet determine how the written interaction acts on those derivatives.

This calculation applies the actual product rule to the local scalar term. It uses the same smooth physical spinor field throughout. Its coordinate directions are ordinary spatial directions on $Q_d$; Taylor coefficients introduce no additional physical dimensions and require no compactification. The result is local and does not choose a global spatial topology.

In a local orthonormal spin frame, write

$$\mathcal E_{\rm sc} =\frac{\kappa}{2}\int(\psi^\dagger\beta\psi)^2d\mu,\qquad \mathcal N(\psi)=\kappa(\psi^\dagger\beta\psi)\beta\psi, \qquad \beta^\dagger=\beta,\quad\beta^2=I. \tag{1}$$

The metric, $\beta$ and $\kappa$ are fixed background data in the variation. The full equation also contains its spatial Dirac operator. No occupied profile or physical mass follows from the algebraic source alone.

Derivative counting and its exact product rule

Choose local real coordinates centred at $q$. For a multiindex $\alpha=(\alpha_1,\ldots,\alpha_d)$, put $|\alpha|=\sum_j\alpha_j$ and $\alpha!=\prod_j\alpha_j!$. The divided Taylor coefficients through order $N$ are

$$c_\alpha=\frac{\partial^\alpha\psi(q)}{\alpha!}, \qquad j_N\psi=(c_\alpha)_{|\alpha|\le N}. \tag{2}$$

The name jet means this finite collection of derivatives. A field of such coefficients is holonomic when the coefficients actually are the derivatives of one field. At one point, arbitrary finite coefficients can be realized by a polynomial; across space they obey derivative compatibility conditions. The coordinate-independent jet definition and its Taylor description are given in Kolář, Michor and Slovák, Sections 12.1–12.5.

Per spinor component, the rank is exactly

$$D_{N,d}=\sum_{k=0}^N\binom{k+d-1}{d-1}=\binom{N+d}{d}. \tag{3}$$

For a fixed smooth real scalar current $J$, denote its divided derivatives by $J_\gamma$. The Leibniz rule gives

$$[j_N(J\psi)]_\alpha =\sum_{\gamma+\delta=\alpha}J_\gamma c_\delta. \tag{4}$$

Thus multiplication induces a truncated convolution matrix

$$[M_N[J]]_{\alpha\delta}= \begin{cases} J_{\alpha-\delta},&\delta_j\le\alpha_j\ \text{for every }j,\\ 0,&\text{otherwise}. \end{cases} \tag{5}$$

Order the indices by total degree. Every diagonal entry is $J_0=J(q)$; every other nonzero entry raises degree. Consequently

$$M_N[J]=J_0I+K,\qquad K^{N+1}=0, \tag{6}$$

and

$$\boxed{ \det(\lambda I-M_N[J])=(\lambda-J_0)^{D_{N,d}}, \qquad \operatorname{tr}(M_N[J]^r)=D_{N,d}J_0^r. } \tag{7}$$

The integer in (3) appears as a multiplicity and in a trace. The sole scalar eigenvalue remains $J_0$. When $J_0\ne0$, the jet multiplier has full rank. Equation (7) is a statement about this particular local operator; it does not bound a separately derived spatial effective action.

Changing to coherent coordinates preserves this distinction

Using unscaled derivatives instead of (2) conjugates (5) by the diagonal matrix with entries $\alpha!$. The usual degree-$N$ homogeneous coherent normalization is another invertible diagonal transformation,

$$S_\alpha=\sqrt{\frac{N!\,\alpha!}{(N-|\alpha|)!}}, \qquad \widetilde c_\alpha=S_\alpha c_\alpha,\qquad \widetilde M=SM_N[J]S^{-1}. \tag{8}$$

It leaves (7) unchanged. It is a change of coefficient coordinates, not a derivation of their kinetic metric.

There is a useful exact complex example. With fixed spinor $u$, amplitude $a$ and real spatial wavevector $k$, let

$$\psi(x)=aue^{ik\cdot x},\qquad c_\alpha=au\frac{(ik)^\alpha}{\alpha!}. \tag{9}$$

At the coordinate origin, (8) turns the scalar coefficient sequence into the homogeneous coherent occupation sequence generated by $(1,ik)$. Its squared coefficient norm is proportional to $(1+|k|^2)^N$, rather than being an independent physical norm of the original field.

Retaining the conjugate phase gives

$$J(x)=|a|^2u^\dagger\beta u,\qquad J_\gamma=0\quad(|\gamma|>0), \tag{10}$$

because the derivative sum in every coordinate contains

$$\sum_{\ell=0}^r(-1)^\ell\binom r\ell=\delta_{r0}. \tag{11}$$

Therefore the complete source satisfies

$$\boxed{j_N\mathcal N(\psi)=\kappa J_0\,\beta\,j_N\psi} \tag{12}$$

at every retained order, before and after (8). This local plane wave is an operator example, not a proposed localized particle. At the assigned six-dimensional cumulative indices:

This comparison holds the physical background fixed; a separate nonlinear branch could have a different current profile.

Include the full variation of the self-consistent current

For the nonlinear source in (1), the current is not fixed. Its actual Taylor coefficients and those of its source are

$$J_\alpha=\sum_{\gamma+\delta=\alpha}c_\gamma^\dagger\beta c_\delta, \qquad \mathcal N_\alpha =\kappa\sum_{\gamma+\delta=\alpha}J_\gamma\beta c_\delta. \tag{13}$$

All $J_\alpha$ are real; conjugate terms pair under $\gamma\leftrightarrow\delta$. Define $v_\alpha=\beta c_\alpha$. The complete complex variation has coefficient matrices

$$\begin{aligned} A_\nu&=\kappa\left[ J_\nu\beta+\sum_{\gamma+\delta=\nu}v_\gamma v_\delta^\dagger \right],\\ B_\nu&=\kappa\sum_{\gamma+\delta=\nu}v_\gamma v_\delta^{\mathsf T},\\ \delta\mathcal N_\alpha &=\sum_{\nu+\delta=\alpha} \left(A_\nu\eta_\delta+B_\nu\overline{\eta_\delta}\right). \end{aligned} \tag{14}$$

This retains every spin polarization and both complex variation blocks. The $B$ term and the additional term in $A$ are the response of the current itself. Freezing the current would omit them.

For any matrices $A,B$, the real matrix of $\eta\mapsto A\eta+B\bar\eta$, using real and imaginary parts, is

$$\mathcal R(A,B)= \begin{pmatrix} \operatorname{Re}(A+B)&-\operatorname{Im}(A-B)\\ \operatorname{Im}(A+B)&\operatorname{Re}(A-B) \end{pmatrix}. \tag{15}$$

Interleave the real and imaginary parts at each derivative index. Equation (14) is block lower triangular, with the same local block $\mathcal R_0=\mathcal R(A_0,B_0)$ on every diagonal. Hence the full real source Jacobian $\mathcal J_N$ obeys

$$\boxed{ \det(\lambda I-\mathcal J_N) =\det(\lambda I-\mathcal R_0)^{D_{N,d}}. } \tag{16}$$

Again the count repeats local response eigenvalues. Spatial derivatives appear in the off-diagonal blocks and can change Jordan structure. Neither this algebraic source Jacobian nor (7) is the full time-evolution generator, which also includes multiplication by $-i$, the spatial Dirac term, and any rotating-frame subtraction.

The local eigenvalues can also be written explicitly. Let the complex $+1$ and $-1$ eigenspaces of $\beta$ have dimensions $r_+$ and $r_-$, both positive, and set

$$Q_0=c_0^\dagger c_0,\qquad S_0=c_0^\dagger\beta c_0.$$

In real coordinates the source derivative is $\kappa[S_0\beta_{\mathbb R}+2v_{\mathbb R}v_{\mathbb R}^{\mathsf T}]$. Its eigenvalues are

$$\begin{array}{c|c} \text{Eigenvalue}&\text{Multiplicity}\\ \hline \kappa S_0&2r_+-1\\ -\kappa S_0&2r_--1\\ \kappa\left(Q_0+\sqrt{Q_0^2+3S_0^2}\right)&1\\ \kappa\left(Q_0-\sqrt{Q_0^2+3S_0^2}\right)&1 \end{array}. \tag{17}$$

To derive the last two values, remove the directions orthogonal to the positive and negative parts of $c_0$. Their squared norms are $(Q_0+S_0)/2$ and $(Q_0-S_0)/2$. On the remaining real two-dimensional span, the trace is $2\kappa Q_0$ and the determinant is $-3\kappa^2S_0^2$. Formula (17) also holds by continuity when one part vanishes. At a scalar-null field, $S_0=0$, the only nonzero local eigenvalue is $2\kappa Q_0$. A sign of $\beta$ or an eigenvalue in (17) is not an assignment of particle versus antiparticle.

Curvature and the physical kinetic norm

On a curved spatial manifold, a connection splits the finite derivative filtration into symmetrized covariant derivatives. The change from ordinary coordinate derivatives to this splitting is triangular and invertible, with the same highest-order symbols. A connection preserving the spinor pairing obeys the covariant product rule. The associated graded operator of scalar multiplication remains $J(q)$ on every degree, so (7) survives. The full source variation transforms with the same fixed linear change of jet coordinates, preserving (16). Curvature does not turn a coordinate transformation into a new scalar coupling.

There is a useful warning about the norm implicit in such a rewriting. If some retained derivative of $J$ is nonzero, $K$ in (6) is a nonzero nilpotent matrix. Then $M_N[J]$ is not diagonalizable and cannot be self-adjoint in any positive definite metric on the independent finite jet fibre. For example, at $d=N=1$ and $J=1+x$,

$$M=\begin{pmatrix}1&0\\1&1\end{pmatrix}. \tag{18}$$

The equation $GM=M^\dagger G$ forces $G_{22}=0$, which excludes a positive definite $G$. Ordinary real multiplication on the physical $L^2$ field space is nevertheless self-adjoint. There is no contradiction: the Taylor map is not a unitary identification of that space with an independent finite coefficient fibre at one point.

The original canonical norm remains

$$Q[\psi]=\int\psi^\dagger\psi\,d\mu. \tag{19}$$

Expressing its field through holonomic derivatives does not replace (19) with an independent sum over all derivative slots. A positive norm that weights all spatial derivatives defines a different, usually Sobolev, metric; its induced operator and constraints must be transformed as well. The canonical normalization calculation explains why the physical kinetic metric must accompany a projected interaction.

There is also a kinetic closure issue. Already for a flat first-order Dirac operator $\mathscr D=-i\sum_j\mathsf a_j\partial_j$,

$$[j_N(\mathscr D\psi)]_\alpha =-i\sum_j\mathsf a_j(\alpha_j+1)c_{\alpha+e_j}. \tag{20}$$

The highest retained derivative needs order $N+1$ of the field. For example, $\psi_1=u$ and $\psi_2=u+\epsilon x_1^{N+1}v$ have identical $N$-jets at zero, but their degree-$N$ Dirac jets differ by $-i\epsilon(N+1)\mathsf a_1v$. Thus a finite jet at one point does not define a closed algebraic kinetic matrix. A spatial PDE on jet fields can retain their derivatives and compatibility equations; a finite oscillator reduction instead needs an actual projection and control of its complementary spatial modes.

Consequence for the mass-operator derivation

The count in (3) is exact. The prolonged local action gives (13) and (14), rather than the auxiliary rank-one contact. Deriving the latter requires an additional physical reduction, such as a specific source pairing and its canonical normalization. The defined graph pairing supplies one conditional action with a simplex coefficient, but its kernel is an additional premise.

The result concerns the written local scalar term and changes of its derivative coordinates. It does not exclude different nonlinear spatial branches, a derived interaction from complementary modes of the same field, or a different master term. It leaves the assigned particle dimensionalities and mode indices unchanged.

Verification

local_current_jet_checks.py compares (13) with independently expanded complex spatial polynomials. It checks the complete real-linear variation, trace identities, both signs of $\beta$, scalar-null fields, and covariance when all spin data are rotated together. The largest explicit real Jacobian has dimension $448$, retaining every derivative through order two in six dimensions for an eight-component spinor.

A nonflat metric-compatible connection is checked in two independent ways: symmetrized ordered covariant differentiation and a matrix exponential along radial lines. Their current jets and full nonlinear variations agree. Exact binomial identities verify the plane-wave cancellation at arbitrary multiindices through the assigned degrees, without constructing millions of coefficient entries. Symbolic differentiation also verifies the kinetic counterexample (20).