Frequency accounting for scalar Born response

Summary: Complete scalar vertices remove the negative-frequency resonant Born channel for positive incoming waves, while a finite-time positive preparation can retain a nonresonant response; the distinction follows from spatial momentum, frequency and both angular contractions.

Positive incoming bands and the complete cubic source

Consider the specified local scalar equation on $\mathbb R^D$, $D\ge2$,

$$i\partial_t\Psi=H(-i\nabla)\Psi +g(\Psi^\dagger\beta\Psi)\beta\Psi, \quad H(p)=\alpha\cdot p+m\beta, \quad E_p=\sqrt{|p|^2+m^2},\quad m\ge0. \tag{1}$$

The matrices retain the complete spatial Clifford module, as in the positive-frequency scalar vertex. Its minimal complex size is $n=2^{\lfloor(D+1)/2\rfloor}$. The spatial directions are ordinary physical coordinates. A specified free mass coefficient $m$ is an input to this comparison. The signed projectors are

$$P_\pm(p)=\frac12\left(I\pm\frac{H(p)}{E_p}\right). \tag{2}$$

For $m=0$, the single point $p=0$ has no defined signed band and is excluded where a projector is used. It has zero continuum measure. Let the complete incoming Fourier spinor satisfy $A(p)=P_+(p)A(p)$, so the free field is

$$\Psi_0(x,t)=\int\frac{d^Dp}{(2\pi)^{D/2}} A(p)e^{ip\cdot x-iE_pt}. \tag{3}$$

The negative-frequency projection is a spectral component of this classical field. Its physical particle interpretation remains open; it is not assigned an antiparticle meaning or a sign of electric charge.

The full negative-output Born coefficient

Build the local scalar before projecting the cubic source. At output momentum $k$, put

$$p=q+r-k,\qquad V(k;q,r)=P_-(k)\beta A(r) \left[A(p)^\dagger\beta A(q)\right], \qquad \delta=E_q+E_r+E_k-E_p. \tag{4}$$

The first Born term in the interaction picture over a time interval $I$ is

$$\boxed{\mathcal B^-_I(k) =-\frac{ig}{(2\pi)^D} \int d^Dq\,d^Dr\,V(k;q,r) \int_I e^{-i\delta t}\,dt.} \tag{5}$$

The conjugated slot contributes $e^{-ip\cdot x+iE_pt}$; the two unconjugated slots contribute $e^{i(q+r)\cdot x-i(E_q+E_r)t}$. Integration over both $q,r$ retains both coherent permutations. Treating only one selected scalar contraction as the whole source would omit terms.

Formula (5) is an explicitly computed cubic response of the same field. It introduces no independent response particle or interaction coefficient. It is not a quantum-fermion scattering matrix.

The massive resonance cannot close

Write $w=-k$, so $p=q+r+w$. For $m>0$, convexity of the relativistic energy gives

$$E_q+E_r+E_w\ge 3\sqrt{|p/3|^2+m^2}=\sqrt{|p|^2+9m^2}. \tag{6}$$

Equality occurs at $q=r=w=p/3$. Therefore

$$\boxed{\delta\ge \sqrt{|p|^2+9m^2}-\sqrt{|p|^2+m^2}>0.} \tag{7}$$

There is no negative-output on-shell cubic resonance among three positive incoming free-frequency factors with this common mass. This is a kinematic statement about the signed source equation, not a statement that its finite-time forcing is zero.

The massless resonance has two vanishing scalar contractions

For $m=0$, let $a=|q|$, $b=|r|$, $c=|w|$, $A_0=a+b+c$, and $P=|p|$. Then $\delta=A_0-P\ge0$ by the triangle inequality. For nonzero $q,r,w$, equality requires all three to point in the same direction as $p$.

The complete scalar projector norms are

$$\begin{aligned} \|P_+(p)\beta P_+(q)\|_{\rm op} &=\sin\frac{\theta_{pq}}2,\\ \|P_-(k)\beta P_+(r)\|_{\rm op} &=\sin\frac{\theta_{wr}}2. \end{aligned} \tag{8}$$

The first is the established positive-band half-angle identity. For the second, conjugating $P_-(k)$ by $\beta$ and compressing to $P_+(r)$ gives squared singular strength $(1+\widehat k\cdot\widehat r)/2 =(1-\widehat w\cdot\widehat r)/2$. Both contractions vanish on the massless resonant geometry. The statement retains every polarization and is independent of a special depth-five assignment.

A bound through the collinear limit

A zero of the numerator alone is insufficient to handle a small frequency denominator. Here its cancellation is quantitative. For $p,q,r,w\ne0$,

$$\begin{aligned} \delta&=\sum_{j=q,r,w}|j| \left(1-\widehat p\cdot\widehat j\right),\\ (A_0+P)\delta&=4\left[ ab\sin^2\frac{\theta_{qr}}2 +ac\sin^2\frac{\theta_{qw}}2 +bc\sin^2\frac{\theta_{rw}}2\right]. \end{aligned} \tag{9}$$

The first identity follows by taking the scalar product of $p=q+r+w$ with $\widehat p$; the second follows by expanding $A_0^2-|q+r+w|^2$. All terms on their right are nonnegative. Thus

$$\sin\frac{\theta_{pq}}2\le\sqrt{\frac{\delta}{2a}}, \qquad \sin\frac{\theta_{wr}}2\le \sqrt{\frac{(A_0+P)\delta}{4bc}}. \tag{10}$$

Combining both complete vertices yields

$$\boxed{\frac{\|V(k;q,r)\|}{\delta} \le\sqrt{\frac{A_0+P}{8abc}} \|A(p)\|\,\|A(q)\|\,\|A(r)\|.} \tag{11}$$

This bound holds off resonance and remains bounded as the nonzero momenta become collinear. It does not discard the output infrared dependence $c^{-1/2}$. For a finite time window, $|\int_0^T e^{-i\delta t}dt|\le2/\delta$; (11) consequently controls the full nonresonant source contribution without a secular collinear pole.

A localized Born scattering limit

For a precise continuum statement, assume $A\in C_c^\infty(\mathbb R^D\setminus\{0\};\mathbb C^n)$, $A=P_+A$, with momentum support $0<\kappa\le|p|\le\Lambda$. This describes smooth spatially decaying free fields, rather than isolated plane waves or a finite periodic regulator.

For the symmetric interval $I=[-T,T]$, (5) contains $2\sin(T\delta)/\delta$. In the massless case, (11) makes $V/\delta$ integrable over the compact $q,r$ support for every fixed $k\ne0$. Values on its measure-zero resonant set can be assigned arbitrarily for this integral. The phase gradient is

$$\nabla_{q,r}\delta= (\widehat q-\widehat p,\widehat r-\widehat p). \tag{12}$$

Its critical set requires $q,r,p$ to be collinear with the fixed $w=-k$, a measure-zero set when $D\ge2$. Away from that set, a local change to the phase as one coordinate gives an integrable phase density. The Riemann–Lebesgue lemma then makes its oscillatory integral tend to zero for every nonzero output $k$.

The output is supported in $|k|\le3\Lambda$, and (11) also supplies a uniform bound proportional to $|k|^{-1/2}$. Its square is locally integrable at zero for $D\ge2$, since the radial integral is proportional to $\int_0^{3\Lambda}\rho^{D-2}d\rho$. Dominated convergence therefore gives

$$\boxed{\lim_{T\to\infty}\|\mathcal B^-_{[-T,T]}\|_{L^2_k}=0.} \tag{13}$$

For $m>0$, the uniform gap (7) on the bounded incoming support makes the quotient bounded. The phase gradient is $(q/E_q-p/E_p,r/E_r-p/E_p)$; its critical set at fixed $k$ is the single possible point $p=q=r=k$. The same argument gives (13).

This proof uses the integrable oscillatory coefficient directly; it does not treat a delta function of the degenerate massless resonance as an ordinary regular hypersurface measure. Equation (13) concerns the explicit Born term. Interpreting it as the derivative of a complete nonlinear incoming-to-outgoing map additionally requires that map and its differentiability.

A finite-time positive preparation is different

For $I=[0,T]$,

$$\int_0^T e^{-i\delta t}dt =\frac{1-e^{-i\delta T}}{i\delta}. \tag{14}$$

Under the same localized assumptions, the oscillatory part vanishes after momentum integration as $T\to\infty$, but the constant part can leave

$$\boxed{\mathcal B^-_{[0,\infty)}(k) =-\frac{g}{(2\pi)^D} \int d^Dq\,d^Dr\,\frac{V(k;q,r)}{\delta}.} \tag{15}$$

For a concrete nonzero source, in two dimensions take $\alpha=(\sigma_x,\sigma_y)$, $\beta=\sigma_z$ and unit positive spinors $\xi_j=(1,e^{i\theta_j})^T/\sqrt2$ along the momenta $p=(0,2)$, $q=(2,1)$, $r=(-1,2)$. At $k=(1,1)$ the two coherent permutations have a combined negative-output norm approximately $0.199709$, with $\delta=2\sqrt5+\sqrt2-2$. Narrow smooth momentum bumps around these separated carriers preserve a nonzero contribution to (15) by continuity; the other generated momenta are distinct. The isolated plane waves themselves are only source controls and do not have the localized scattering limit (13).

This is a nonresonant correction associated with the finite initial preparation. Its existence is compatible with (13). The localized encounter simulations prepare positive free-frequency packets at a finite time; their tails already overlap. Positive preparation therefore does not justify deleting subsequently generated negative-frequency components or assigning their norm a higher perturbative order without further work. The initial-data prescription also matters for the retarded local response about its specified background; that separate spectrum is not the free spectrum used above.

Verification and mathematical context

scalar_born_response.py implements the full projectors, cubic vertex, stable angular detuning and both time-window kernels. scalar_born_checks.py runs 1,205 checks of full Clifford modules at every spatial dimension from two through ten, including nearly collinear bands and both massive and massless detuning identities. An independent spatial Fourier extraction retains both coherent permutations. Direct forced evolution of the complete signed Dirac mode verifies its nonzero finite-time response and its normalization. Common physical rotations preserve the full vector source.

The broader nonlinear theory already exploits related scalar null structure. Bournaveas and Candy, Theorem 1.1 prove small-data global existence and scattering for the massless cubic Soler and Thirring equations at the critical homogeneous Sobolev regularity in two and three spatial dimensions. Their Remark 1.4 explains the inhomogeneous version using conserved $L^2$ norm. That theorem is not a global-existence claim for the higher-dimensional finite-amplitude preparations considered here.

These frequency restrictions constrain a specified classical scalar term. They supply no charge assignment, bound-particle identification, or observed coupling rate. The conserved spatial charge question remains a separate physical requirement.