Time ordering of spatial compatibility

Summary: The leading change of the prescribed spatial contact is a Poisson bracket with the second time-ordered source generator, identifying a reciprocal feedback contribution that a single integrated overlap omits.

Keep the full spatial Hamiltonian at each stage

The localized compatibility functional compares the same free encounter for the incoming field and the fully evolved outgoing field after removing known free propagation. Its leading scattering variation preserves that functional. The next term can be derived from the same scalar source without adding an alignment rule.

Use the local scalar Dirac equation,

$$i\partial_t\Psi=D\Psi+g(\Psi^\dagger\beta\Psi)\beta\Psi, \qquad U(t)=e^{-iDt},\qquad \psi_I(t)=U(-t)\Psi(t). \tag{1}$$

For a canonical field argument $\psi$, define

$$\chi_t=U(t)\psi,\quad \sigma_t=\chi_t^\dagger\beta\chi_t, \qquad h_t[\psi]=\frac12\int\sigma_t^2\,d^Dx. \tag{2}$$

These are full spatial functionals; both complex patterns enter the same local scalar. The interaction-picture equation is Hamiltonian with $g h_t$. Its explicit time dependence comes from the free spatial operator, even though (1) is autonomous.

Use the positive spatial inner product and gradient convention $\delta F=2\operatorname{Re}\langle\nabla_*F,\delta\psi\rangle$. The canonical Poisson bracket is

$$\boxed{\{F,G\}=2\operatorname{Im} \langle\nabla_*F,\nabla_*G\rangle,\qquad \dot F=\{F,G\}\quad\hbox{when }\dot\psi=-i\nabla_*G.} \tag{3}$$

All formulas below are finite-order, fixed-duration expansions where regular solutions depend smoothly on $g$. They hold directly in a finite canonical spatial regulator. A continuum use additionally needs sufficient regularity and control of the displayed derivatives and integrals; no convergence of an infinite Magnus series is asserted.

The first two source generators

Let $T>0$ be fixed. Define

$$\boxed{ K_1=\int_0^T h_t\,dt=\frac12\mathcal C_T, \qquad K_2=\frac12\int_0^Tdt\int_0^t ds\,\{h_t,h_s\}. } \tag{4}$$

For any sufficiently differentiable real functional $F$, the complete interaction-picture map gives

$$\begin{aligned} F[\psi_I(T)]={}&F[\psi_I(0)]+g\{F,K_1\}\\ &+g^2\left(\{F,K_2\} +\frac12\{\{F,K_1\},K_1\}\right)+O(g^3), \end{aligned} \tag{5}$$

with the brackets on the right evaluated at the incoming field. For the sign convention, write $L_G F=\{F,G\}$. Two stages, $h_s$ first and $h_t$ second, pull back observables in the order $e^{gL_{h_s}}e^{gL_{h_t}}$. The Jacobi identity gives $[L_{h_s},L_{h_t}]=L_{\{h_t,h_s\}}$. Expanding that ordered product through second order yields (4)–(5). Integrating over all ordered pairs gives the displayed time triangle. This is the Hamiltonian form of the standard chronological expansion; see Blanes, Casas, Oteo and Ros, section 3.4.1.

Taking $F=\mathcal C_T=2K_1$ eliminates both the first-order term and the repeated first-generator bracket:

$$\boxed{\Delta\mathcal C_T =g^2\{\mathcal C_T,K_2\}[\psi_{\rm in}]+O(g^3).} \tag{6}$$

Thus the first contact change is controlled by how the full spatial Hamiltonians at different encounter stages fail to commute. The integral of their separate contact strengths is insufficient to determine it. The sign of (6) is not fixed by the nonnegativity of $\mathcal C_T$.

If $\{h_t,h_s\}=0$ as functionals for every pair of stages on the relevant domain, then $\{\mathcal C_T,h_t\}=0$ and the full interaction-picture evolution conserves $\mathcal C_T$ exactly. By contrast, a zero numerical value of $K_2$ at one incoming field does not ensure a zero gradient or a zero bracket in (6).

The coefficient retains both complete complex variations

The full source gradient and its real-linear derivative are

$$\begin{aligned} f_t=\nabla_*h_t &=U(-t)[\sigma_t\beta\chi_t],\\ H_t\eta=(\mathrm d f_t)_\psi[\eta] &=U(-t)\left[ 2\operatorname{Re}(\chi_t^\dagger\beta U(t)\eta)\,\beta\chi_t +\sigma_t\beta U(t)\eta\right]. \end{aligned} \tag{7}$$

The inner product inside the real part in (7) is pointwise; the outer source remains a complete spatial field. In a Galerkin implementation, the local product is followed by the same orthogonal spatial projection as the original energy variation.

Since $H_t$ is symmetric for the real canonical inner product,

$$\begin{aligned} K_2&=\int_0^Tdt\int_0^t ds\, \operatorname{Im}\langle f_t,f_s\rangle,\\ \nabla_*K_2 &=\frac12\int_0^Tdt\int_0^t ds\, \left[H_t(-if_s)+H_s(if_t)\right]. \end{aligned} \tag{8}$$

For example, varying the integrand of $\{h_t,h_s\}$ gives $2\operatorname{Im}(\langle H_t\eta,f_s\rangle+ \langle f_t,H_s\eta\rangle)$; moving the two real-symmetric Hessians to the other slot gives (8). With $F_1=\int_0^T f_tdt$, the coefficient in (6) is

$$\boxed{\{\mathcal C_T,K_2\} =4\operatorname{Im}\langle F_1,\nabla_*K_2\rangle.} \tag{9}$$

Dropping the conjugate-dependent part of (7), holding one pattern fixed, or replacing its local scalar by an average would change (8)–(9). No independent coefficient has been introduced: $g$ is still the single scalar coefficient of (1).

An independent full-field expansion

Iterating the interaction-picture differential equation directly gives

$$\psi_I(T)=\psi_{\rm in}-igF_1+g^2 B_2+O(g^3), \quad B_2=-i\int_0^Tdt\int_0^t ds\,H_t(-if_s). \tag{10}$$

The generator formula instead gives

$$B_2=-i\nabla_*K_2-\frac{i}{2} \int_0^T H_t(-iF_1)dt. \tag{11}$$

Splitting the square time integral in (11) into its two triangles and using (8) reproduces (10). This provides a check on the source ordering and the factors of two, without interpreting the generator as a new physical field.

Reversing the prescribed sequence $h_t\mapsto h_{T-t}$ leaves $K_1$ unchanged and sends $K_2\mapsto-K_2$. It therefore reverses the coefficient (6). This is a control on a sequence of Hamiltonians, not a claim that freely departing spatial packets reverse themselves or naturally repeat an encounter.

Compute the coefficient by direct coupling variations

The same coefficient can be calculated without the double time integral. Expand the physical field at fixed preparation as $\Psi_g=\chi+g a+g^2 b+O(g^3)$, with $\chi=U(t)\psi_{\rm in}$. Writing $N(\chi)=(\chi^\dagger\beta\chi)\beta\chi$, direct variation of the complete spatial equation gives

$$\begin{aligned} i\partial_t a&=Da+N(\chi),&a(0)&=0,\\ i\partial_t b&=Db+(\mathrm dN)_\chi[a],&b(0)&=0. \end{aligned} \tag{12}$$

The source derivative is the full real-linear expression in (7), before conjugation by the free propagators. It includes the response of both spatial patterns. In the interaction picture, $a_I(T)=-iF_1$ and $b_I(T)=B_2$. Let $\mathcal H_C=2\int_0^T H_tdt$ be the real Hessian of the prescribed contact. Taylor expansion of that same functional gives

$$\boxed{c_2=2\operatorname{Re}\langle\nabla_*\mathcal C_T,b_I(T)\rangle +\operatorname{Re}\langle a_I(T),\mathcal H_C a_I(T)\rangle, \qquad \Delta\mathcal C_T=g^2 c_2+O(g^3).} \tag{13}$$

Equations (6) and (13) give the same coefficient using different calculations. The direct method integrates two inhomogeneous spatial variations and one single time quadrature. It does not fit a coefficient to completed nonlinear collisions.

spatial_packet_variations.py implements (12)–(13), with timestamped results and optional variation fields. Its local Hessian differentiates every quadrature component of the original scalar energy. spatial_packet_variation_checks.py runs 28 checks, including independent cubic polarization, scalar energy variation, both norm coefficients, comparison with the ordered generator and full nonlinear evolution. The same checks include positive renormalization of both complete outgoing patterns.

Verification and physical scope

packet_time_order.py evaluates the two generators and both full-field expressions using the normalized axial spatial equation. For its homogeneous cubic source, the Hessian also has the exact polarization identity

$$H_t\eta=\frac{f_t(\psi+\eta)-f_t(\psi-\eta)}2-f_t(\eta). \tag{14}$$

packet_time_order_checks.py runs 50 checks. It checks (14) by independent directional differentiation, verifies real Hessian symmetry and the scalar variation of $K_2$, refines the time quadrature and checks source-order reversal. Complete nonlinear spatial evolution at dimensions three and five approaches (10) with cubic field error and approaches the contact coefficient (9) under small-coupling refinement. A frozen-transport control conserves the same contact under its full spatial nonlinear source.

The result explains a conditional feedback contribution in the stated comparison protocol. It does not select occupied bound-particle patterns, fix their physical sizes, assign observed interaction channels, or imply a universal tendency toward improved compatibility.