Spatial derivative compatibility and the current hierarchy

The cumulative derivative count has the desired rank,

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

This note tests whether actual derivatives of a localized field have the coherent structure used in the reconstructed current action. The answer depends on their compatibility across physical space. It is not determined by counting coefficient slots at one point.

All coordinates below are ordinary spatial coordinates of one sector. Derivative coefficients and the harmonic representations used to compute their contractions add no spatial dimensions. The flat and Gaussian calculations are conditional spatial controls, not an identified particle geometry or a replacement for the assigned particle indices.

A coherent derivative ray is strongly constrained

Let psi be a smooth vector or matrix field on a connected region of R^d. It may include every complex spinor component and partner column. Choose one constant derivative length L_j and define the homogeneous normalization of its ordinary derivatives through order N by

$$(J_N\psi)_\alpha= \sqrt{\frac{N!}{(N-|\alpha|)!\,\alpha!}} L_j^{|\alpha|}\partial^\alpha\psi, \qquad |\alpha|\le N. \tag{2}$$

The omitted homogeneous occupation is alpha_0 = N-|alpha|. These coefficients are a field of jets only when they obey the derivative compatibility equations. The standard jet definition is reviewed in Kolár, Michor and Slovák, Natural Operations in Differential Geometry, Sections 12.1–12.5.[1] The following conclusions follow directly from the product rule and do not require an assumed interaction.

Suppose N is at least two and J_N psi is an exact coherent tensor Phi(x) tensor z(x)^N at every point of an open set where psi is nonzero. The zeroth coefficient implies z_0 is nonzero. Put xi_a = z_a/(L_j z_0). Comparing the first and second coefficients gives

$$\partial_a\psi=\xi_a\psi,\qquad \partial_a\partial_b\psi=\xi_a\xi_b\psi. \tag{3}$$

Differentiating the first equation and using the second yields (partial_a xi_b) psi = 0. Hence every xi_a is constant and

$$\boxed{\psi(x)=e^{\xi\cdot x}\psi_*} \tag{4}$$

on each connected nonzero region. This includes complex phase gradients and every spinor component. A smooth nonzero solution cannot end on a finite boundary of that region, since (4) has a nonzero finite limit there. On all of R^d it is therefore a single exponential. No nonzero such field has finite ordinary L2 norm.

This result concerns an exact coherent ray at every spatial point. An arbitrary finite jet at one point remains possible. Approximate coherence, independent mode-bundle fields, and other physical carriers are different possibilities and are not excluded by (4).

A Hermitian connection does not remove the norm condition

On a Riemannian spatial manifold, use symmetrized covariant derivatives of a Hermitian vector bundle. The corresponding coherent conditions are

$$\nabla_a\psi=\xi_a\psi,\qquad \nabla_{(a}\nabla_{b)}\psi=\xi_a\xi_b\psi. \tag{5}$$

They imply nabla_(a xi_b) = 0. If rho = psi^dagger psi, metric compatibility gives d log rho = 2 Re xi, so

$$\boxed{\nabla_a\nabla_b\log\rho=0.} \tag{6}$$

Thus grad log rho is parallel. On a closed connected manifold rho is constant. On a complete infinite-volume manifold it cannot give a nonzero finite-norm field: if the gradient vanishes, rho is constant; if it does not, its complete parallel flow preserves spatial volume and changes rho exponentially along the flow. Integrating disjoint translated neighborhoods makes the norm diverge. Equation (6) also prevents a nonzero region from ending at a finite-distance zero.

The antisymmetric equation retains curvature,

$$F_{ab}\psi=(\nabla_a\xi_b-\nabla_b\xi_a)\psi. \tag{7}$$

It cannot be discarded to declare the connection flat. The localization condition (6), however, already follows from the symmetric derivatives and the positive fibre norm. This is a statement about the displayed Hermitian jet construction, not a choice of global sector topology.

Oscillator derivatives supply a different, localized construction

Ordinary oscillator lowering operators on R^d are

$$a_j=\frac1{\sqrt2}\left(\frac{x_j}{L}+L\partial_j\right), \qquad [a_i,a_j^\dagger]=\delta_{ij}. \tag{8}$$

Their common eigenfield has the localized form

$$\psi_\zeta(x)=C\exp\left[-\frac{|x|^2}{2L^2} +\frac{\sqrt2\,\zeta\cdot x}{L}\right]\chi, \qquad a_j\psi_\zeta=\zeta_j\psi_\zeta. \tag{9}$$

Replacing L_j partial_j in (2) by a_j therefore gives an exact coherent coefficient sequence. These are oscillator derivatives with an explicit common spatial scale; they are not ordinary covariant derivatives of a Hermitian connection. The real x_j/L term changes the spatial operator. Its confining field origin must still be derived from the same action. The Hermite differentiation and recurrence identities provide independent checks of this elementary construction.[2]

Its canonical norm also needs to be retained. If an independent canonical term is assigned to every lifted coefficient and then pulled back to psi, the induced positive kinetic operator is

$$Z_N=J_N^\dagger J_N =\sum_{k=0}^N\binom Nk\widehat n^{\underline k}, \qquad \widehat n=\sum_j a_j^\dagger a_j, \tag{10}$$

where n-underlined-k is the falling factorial. To prove (10), expand the normal-ordered k-fold power of the total number operator: the coefficient of (a^dagger)^alpha a^alpha is k!/alpha!. Combining it with binomial(N,k) gives exactly (2)'s squared coefficient. On (9), the integrated lifted norm is (1+|zeta|^2)^N times the original field norm. On the Gaussian ground field, every lowering derivative vanishes, and that factor is one for every cutoff N.

For ordinary derivatives, the corresponding integrated operator is instead

$$Z_N^{\rm ordinary}=(1-L_j^2\nabla^2)^N, \tag{11}$$

as follows from integration by parts and the multinomial theorem. Both statements assume decaying fields or compatible boundary domains. A mere rewriting of the original field action does not add the independent coefficient kinetic terms in the first place. Equations (10)–(11) identify the norm that such an added lift would require.

The cutoff is still a description choice for these common eigenfields. Changing N in their derivative list does not make different physical stationary fields or derive the assigned particle mass-frequencies.

Exact reduced currents of ordinary Gaussian derivatives

A Gaussian supplies a useful fully spatial comparison to the exact-ray assumption. Take psi(x) = exp[-|x|^2/(2L^2)] chi and evaluate (2) at its center. Define lambda = L_j^2/L^2. Its scalar derivative moments vanish at odd order, while

$$L_j^{2|a|}\partial^{2a}\psi(0) =(-\lambda)^{|a|}\prod_i\frac{(2a_i)!}{2^{a_i}a_i!}\,\chi. \tag{12}$$

These are actual field derivatives. The squared coefficient norm per unit central norm is

$$Z_{N,d}(\lambda)= \sum_{j=0}^{\lfloor N/2\rfloor} \frac{N!}{(N-2j)!}\frac{(d/2)_j}{j!}\lambda^{2j}. \tag{13}$$

The generating function product_i (1-t)^(-1/2) = (1-t)^(-d/2) sums the even-derivative weights and proves (13). This pointwise quantity is distinct from the integrated operator in (11).

Split the normalized symmetric N-tensor into k retained and l = N-k discarded indices. Its complete partial trace can be evaluated using ordinary O(d) harmonic polynomials. Let ell be their harmonic degree,

$$g_{\ell,d}=\binom{\ell+d-1}{d-1} -\binom{\ell+d-3}{d-1}, \tag{14}$$

where the second term is zero for ell below two. For each ell from zero through min(k,l), the normalized Schmidt matrix has entries

$$\boxed{ (M_\ell)_{ab}= \frac{(-\lambda)^{\ell+a+b}}{\sqrt{Z_{N,d}}} \sqrt{\frac{k!\,l!}{(k-\ell-2a)!\,(l-\ell-2b)!}} \frac{(s)_{a+b}}{\sqrt{a!\,b!\,(s)_a(s)_b}}, \quad s=\ell+d/2, } \tag{15}$$

with 0 <= a <= floor((k-ell)/2) and the analogous range for b. Each singular value of M_ell occurs g_(ell,d) times. These blocks retain the entire angular decomposition, rather than sampling rays.

For an explicit derivation, use the Bargmann polynomial of the derivative moments and split its variables into z and w. Its cross factor is exp(-lambda z dot w). The harmonic decomposition of its degree-n term uses the squared norm 4^a a!(ell+d/2)_a of (z dot z)^a times a unit harmonic. Multiplication by the separate Gaussian factors leaves the finite sum

$$\sum_{j=0}^{\min(a,b)} \frac1{(a-j)!(b-j)!j!(s)_j} =\frac{(s)_{a+b}}{a!b!(s)_a(s)_b}. \tag{16}$$

This terminating Vandermonde identity gives (15), including its sign and normalization. In particular,

$$\operatorname{Tr}\rho_k=\sum_\ell g_{\ell,d}\|M_\ell\|_F^2=1, \qquad P_k:=\operatorname{Tr}\rho_k^2 =\sum_\ell g_{\ell,d}\operatorname{Tr}[(M_\ell M_\ell^\dagger)^2]. \tag{17}$$

P_k equals P_(N-k), as required for a pure full coefficient state. For example, at N=2 and k=1 the Schmidt spectrum consists of 1/(1+d lambda^2) once and lambda^2/(1+d lambda^2) repeated d times. At positive lambda it is not a pure ray.

What the current-root action actually sees

For an isotropic partner field times this scalar coefficient state, the normalized current metric of the reconstructed action gives

$$\frac{F_q(R_k)}{Q}=\sqrt{1-q+qP_k},\qquad \frac{U_{\rm root}}{\omega_0 Q} =A_{N,d}:=\sum_{k=0}^N h_{k,d}\sqrt{1-q+qP_k}. \tag{18}$$

Here Q is the local coefficient norm for this comparison. At q>0, an entangled partial current lowers its factor from one. The endpoint currents k=0,N still have P_k=1. The following central-profile values use the previously examined q=1/(2^(d/2)+1), one common lambda=1, and the fixed derivative cutoffs from the assigned mass counts:

These are current-energy coefficients at a specified spatial point. They are not particle frequencies: the full current source need not be parallel to the jet, the field has not solved a stationary equation, and the holonomic kinetic constraint has not been projected into that action. Independent complete small-cutoff calculations explicitly find a nonzero source component orthogonal to generic Gaussian jets. There is a useful exception: at N=2 and lambda=1, rho_1=I/(d+1), and the central source is parallel to the isotropic paired tensor. This is a local symmetry property, not a full spatial stationary solution. Treating (18) generally as an eigenfrequency would repeat the trace-versus-frequency error.

There is also a useful analytic small-length expansion. At fixed N,d,

$$P_k=1-2d\,k(N-k)\lambda^2+O(\lambda^4). \tag{19}$$

The leading entangled Schmidt block has ell=1, with singular value -lambda sqrt(k(N-k)) in each of d directions. The other first-order deformations are local to one side of the split. They do not change its purity at this order. Using the exact Hockey-Stick moment

$$\frac1{D_{N,d}}\sum_{k=0}^N h_{k,d}k(N-k) =\frac{dN(N-1)}{(d+1)(d+2)} \tag{20}$$

in (18) yields

$$\boxed{\frac{A_{N,d}}{D_{N,d}} =1-\frac{q d^2N(N-1)}{(d+1)(d+2)}\lambda^2 +O(\lambda^4).} \tag{21}$$

At depth six and q=1/9 the coefficient is N(N-1)/14. The two assigned cutoffs therefore have different corrections under the same derivative metric. Letting lambda approach zero restores coherence by concentrating the actual coefficient norm into its zeroth derivative; it does not explain occupation of the counted derivative slots.

The constructive outcome is a complete, efficiently computable test of a physical derivative carrier, including all its reduced currents and its induced kinetic norm. A successful mass derivation must supply a carrier and projected action that meet these conditions together. The existence of the correct rank or a local coherent coordinate system alone does not meet them.

Verification and sources

spatial_jet_reductions.py evaluates (13), (15), (17) and (18) without large dense coefficient spaces. The companion checker compares them against independent explicit derivative tensors and every reduced eigenvalue at smaller cutoffs in depths two, three, six and ten. It also checks exact rational contractions (16), complex spatial differentiation, the induced oscillator metric, the complete small-cutoff root source, and the expansions at the unchanged assigned cutoffs. The dated reports preserve the exact executed sources and their scope.

[1] Ivan Kolár, Peter W. Michor and Jan Slovák, Natural Operations in Differential Geometry, Chapter IV, Section 12. This supplies standard jet definitions and coordinate conventions; the compatibility and current calculations are derived above.

[2] NIST Digital Library of Mathematical Functions, Recurrence Relations and Derivatives, Section 18.9. Only the Hermite polynomial identities are used as mathematical checks; no physical mechanism is imported from that reference.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.