Nested sector reciprocity

Summary: Compatible spatial marginals and their dual pullbacks give reciprocal coupling equations on arbitrary nested sectors, with exact composition along a chain.

Spatial maps and measures

All sectors are ordinary macroscopic spatial submanifolds in one nested chain within the infinite physical manifold. The calculation uses any finite pair $Q_a\subset Q_b$, not a particular particle. Inclusion alone supplies neither a global product nor a retraction. Assume a specified map $r_{ba}:Q_b\to Q_a$ that agrees with the inclusion where required, or work in a product neighborhood on which such a map exists.

Specify measures through a disintegration

$$d\mu_b(x,y)=d\mu_a(x)\,d\nu_x(y).$$

The fibre measure $d\nu_x$ includes the geometric Jacobian. It is not automatically a probability measure. For integrable real scalar sources $j_b$, define the marginal and the pullback of a scalar test field $f_a$:

$$(R_{ba}j_b)(x)=\int_{r_{ba}^{-1}(x)}j_b(x,y)\,d\nu_x(y), \qquad (R_{ba}^{*}f_a)(x,y)=f_a(x).$$

Fubini's theorem gives the duality identity

$$\int_{Q_a}f_a R_{ba}j_b\,d\mu_a =\int_{Q_b}(R_{ba}^{*}f_a)j_b\,d\mu_b. \tag{1}$$

Here the star denotes the dual map in this integral pairing. On an infinite fibre, the pullback of a nonzero function need not be in $L^2(Q_b)$. Equation (1) remains valid for an $L^1$ source and bounded test field; calling it a bounded Hilbert-space adjoint requires additional domain assumptions. On finite fibres with uniformly bounded volume, the corresponding $L^2$ maps are bounded.

For spinor or tensor sources, first specify how their indices are transported to a common bundle and contracted. A bare integral of components in unrelated frames is not a covariant marginal. Those linear transport and polarization maps must be retained in $R$ and its dual.

One interaction, two reciprocal responses

Let $K_a$ be a fixed real spatial kernel on the shared sector:

$$(K_a u)(x)=\int_{Q_a}K_a(x,x')u(x')\,d\mu_a(x').$$

Its transpose in the measure pairing has kernel $K_a(x',x)$. Choose the sign and coefficient by writing the interaction energy

$$U_{ab}=g\langle j_a,K_a R_{ba}j_b\rangle_a, \qquad S_{\rm int}=-\int U_{ab}\,dt.$$

Vary the two sources independently while keeping the geometry and kernel fixed. The result is

$$V_a=\frac{\delta U_{ab}}{\delta j_a}=gK_a R_{ba}j_b, \qquad V_b=\frac{\delta U_{ab}}{\delta j_b} =gR_{ba}^{*}K_a^{*}j_a. \tag{2}$$

Thus the upper-to-lower response is a marginal followed by the kernel. The reverse response is the transpose kernel followed by a cylindrical pullback. These maps generally have different domains and different matrix entries; reciprocity is their dual relation:

$$\langle \delta j_a,gK_aR_{ba}\delta j_b\rangle_a =\langle gR_{ba}^{*}K_a^{*}\delta j_a,\delta j_b\rangle_b. \tag{3}$$

If a specified positive scalar model has $j_d=|\psi_d|^2$ and canonical time derivative, varying $\psi_d^*$ gives

$$i\partial_t\psi_a=(H_a+V_a)\psi_a,\qquad i\partial_t\psi_b=(H_b+V_b)\psi_b.$$

Real $V_d$ and self-adjoint $H_d$ preserve each norm. For static geometry and kernels, the complete two-field Hamiltonian preserves total energy. This density coupling can exchange energy while preserving the number in each mode family. Particle-changing vertices are additional interaction structures. A Dirac scalar source $\bar\psi\psi$ instead requires its own adjoint and projector variation, as derived in Projected bilocal variation.

Composition through intermediate sectors

For $Q_a\subset Q_b\subset Q_c$, assume compatible projections $r_{ca}=r_{ba}\circ r_{cb}$ and their corresponding disintegrations. Iterated integration proves

$$R_{ca}=R_{ba}R_{cb},\qquad R_{ca}^{*}=R_{cb}^{*}R_{ba}^{*}. \tag{4}$$

This is an exact consistency requirement for direct and staged spatial reduction. Polarization maps obey the same composition only if they intertwine compatibly; projecting out an intermediate component can discard a coupling that a direct map retains.

Equation (4) does not multiply two interaction constants together. Reducing a source through an intermediate sector is not the same physical process as propagating through an intermediate dynamical mode. The latter includes that mode's propagator and allowed vertices.

Normalization and dimensional dilution

For a normalized nonnegative spatial density,

$$\int_{Q_a}R_{ba}\rho_b\,d\mu_a=\int_{Q_b}\rho_b\,d\mu_b=1.$$

In a product chart, $\rho_b(x,y)=p(x)f(y)$ with $\int f\,d\nu=1$ gives $R_{ba}\rho_b=p$. Any containing depth admitting this preparation therefore gives the same local contact $g\int \rho_a p\,d\mu_a$. Equal contact here follows from the same marginal and vertex, not from an assumed probability of hitting a thin sheet.

The normalization of a cylindrical field is a separate issue. A probability-normalized extension through a uniform box of volume $V_F$ has density $\rho_a/V_F$, and its overlap carries $1/V_F$. One cannot cancel this factor without specifying the source, flux, or coupling convention in the action. Changes to the marginal itself can also change the overlap even while its integral stays one.

Weighted discrete verification

For positive quadrature matrices $W_a,W_b$ and sampled marginal $R$, the dual acting on sampled test fields is

$$R^*=W_b^{-1}R^{\mathsf T}W_a.$$

This formula preserves (1), including unequal cell volumes and position-dependent fibre weights. It also makes (3) testable by independent finite differences of the scalar energy. Omitting the weights can falsely break reciprocity.

generalized_coupling_checks.py checks these identities, chain composition, and a reciprocal scalar evolution with unequal spatial weights.

Finite oscillator realization

Oscillator source marginals gives an explicit finite matrix for this spatial integration, with exact weighted Hockey-Stick singular strengths and direct chain composition. Its bilinear-current calculation retains the full normal-mode contraction before applying the source map, including the different width and degree of the resulting current.

Projective spatial realization

Related

• Generalized sector coupling — organizes the common interaction calculation across arbitrary particle sectors.
• Canonical coupling normalization — converts these responses and coherent vertices to normalized mode coordinates.
• Nested sector dynamics — explains the extra intertwining condition for inherited evolution.