Not Every Breakdown Is Quantum

This is a guided reading of a PHY204 term paper written with Xu Ya-Xuan and substantially revised for public release. Both authors approved the revision. The manuscript states where computational and generative-AI tools assisted; the authors checked the derivations, citations, code, and claims.

Suppose a complicated theory and its simpler approximation disagree. What has failed? The approximation might have been truncated too early. The underlying classical theory might have left its domain of validity. Or the numerical method might simply be wrong. Calling all three possibilities “breakdown” hides the question we actually need to answer.

Radiation reaction gives a sharp example. The Abraham–Lorentz (AL) equation contains an unphysical runaway solution. The Landau–Lifshitz (LL) equation removes that direction by reduction of order. Meanwhile, sufficiently strong fields make photon recoil and stochastic emission important, so any deterministic classical equation becomes questionable. Those are different limits, governed by different dimensionless parameters.

This article builds the smallest model in which we can measure the first limit exactly and keep it separate from the second.

Start with the singular limit

In one nonrelativistic spatial dimension, write the external force per unit mass as $f(t)$ . The AL equation is

a(t)=f(t)+\tau_e\dot a(t),

where $\tau_e\simeq6.26\times10^{-24}\,\mathrm{s}$ for an electron. It looks like a small correction because $\tau_e$ is tiny. But rearranging gives

\tau_e\dot a-a=-f.

The small number multiplies the highest derivative. Setting $\tau_e=0$ lowers the order of the differential equation and removes the initial acceleration as independent data. That is a singular perturbation, not an ordinary small change to a stable initial-value problem.

With $f=0$ , the homogeneous equation has

a_h(t)=C e^{t/\tau_e}.

Unless $C$ is exactly zero, a microscopic error grows exponentially. You might expect to pin $C$ down by choosing an initial acceleration, the way an initial-value problem usually works. Here the condition comes from the other end of the problem instead: we demand that the acceleration vanish after the force has disappeared, which selects the non-runaway branch.

Deriving the exact physical branch

Multiply the equation by the integrating factor $e^{-t/\tau_e}$ :

\frac{d}{dt}\left(e^{-t/\tau_e}a(t)\right) =-\frac{1}{\tau_e}e^{-t/\tau_e}f(t).

Integrate from the present time $t$ to a late time $R$ , then take $R\rightarrow\infty$ on the non-runaway branch. The boundary term at $R$ vanishes and we obtain

a_{\mathrm{AL}}(t)=\frac{1}{\tau_e}\int_t^\infty e^{-(s-t)/\tau_e}f(s)\,ds =\int_0^\infty e^{-u}f(t+\tau_e u)\,du.

The exponential kernel is positive, has unit mass, and has width $\tau_e$ . It averages the force over a very short interval—but that interval points into the future. The selected branch is therefore pre-accelerating and formally acausal: the particle starts moving before the force reaches it. Its tiny timescale does not turn it into a causal law; it only makes the acausal interval extraordinarily difficult to resolve experimentally.

Expand the shifted force:

f(t+\tau_e u)=\sum_{n=0}^{\infty} \frac{(\tau_e u)^n}{n!}f^{(n)}(t).

Since $\int_0^\infty e^{-u}u^n\,du=n!$ , the branch has the formal derivative series

a_{\mathrm{AL}}=f+\tau_e\dot f+\tau_e^2\ddot f+\cdots.

Truncating after the first derivative gives the LL approximation in this model:

a_{\mathrm{LL}}=f+\tau_e\dot f.

This calculation explains both the usefulness and the danger of reduction of order. It replaces an unstable higher-order evolution with a local expression, but its accuracy depends on how quickly the force varies.

A Gaussian makes the comparison countable

Choose

f(t)=f_0e^{-t^2/(2T^2)},\qquad x=\frac{t}{T},\qquad \epsilon=\frac{\tau_e}{T}.

After dividing acceleration by $f_0$ , every profile depends only on $x$ and $\epsilon$ :

\frac{a_{\mathrm{AL}}}{f_0}=\int_0^\infty e^{-u} e^{-(x+\epsilon u)^2/2}\,du, \qquad \frac{a_{\mathrm{LL}}}{f_0}=e^{-x^2/2}(1-\epsilon x).

$\epsilon$ is a timescale ratio. A slow pulse has $\epsilon\ll1$ and a rapidly varying stress test approaches $\epsilon=1$ . Move the slider below. The graph recomputes the future integral rather than interpolating stored curves.

Interactive benchmark

Slow the pulse down; watch LL converge

epsilon = 0.20

Pulse-variation parameterslow, 0.02stress test, 1.00

Exact normalized RMS error3.340e-2

Small-epsilon approximation3.326e-2

At small epsilon the two acceleration curves become visually indistinguishable, while the reported error still resolves their difference. Near one, the truncated asymptotic expression is no longer a useful numerical estimate.

The displacement of the exact peak toward negative $x$ is the pre-acceleration made visible. LL reproduces the first-order displacement. What remains is a second-order shape error.

An exact error law, not just a fitted slope

Define the normalized RMS discrepancy

D_{\mathrm{rms}}^2= \frac{\int_{-\infty}^{\infty}|a_{\mathrm{LL}}-a_{\mathrm{AL}}|^2\,dx} {\int_{-\infty}^{\infty}|a_{\mathrm{AL}}|^2\,dx}.

A log–log plot can suggest a power law, but it does not prove one. Fourier space exposes the structure exactly. With $\widehat h(\omega)=\int h(x)e^{-i\omega x}\,dx$ , the differential equation and LL truncation give

\widehat a_{\mathrm{AL}}= \frac{\widehat g}{1-i\epsilon\omega},\qquad \widehat a_{\mathrm{LL}}=(1+i\epsilon\omega)\widehat g,

where $g(x)=e^{-x^2/2}$ . Their difference is

\widehat a_{\mathrm{LL}}-\widehat a_{\mathrm{AL}} =-\frac{\epsilon^2\omega^2}{1-i\epsilon\omega}\widehat g.

Parseval’s identity therefore yields, for every $\epsilon>0$ ,

D_{\mathrm{rms}}^2= \frac{\displaystyle\int_{-\infty}^{\infty} \frac{\epsilon^4\omega^4}{1+\epsilon^2\omega^2}e^{-\omega^2}\,d\omega} {\displaystyle\int_{-\infty}^{\infty} \frac{1}{1+\epsilon^2\omega^2}e^{-\omega^2}\,d\omega}.

Introduce

J(\epsilon)=\int_{-\infty}^{\infty} \frac{e^{-\omega^2}}{1+\epsilon^2\omega^2}\,d\omega =\frac{\pi}{\epsilon}e^{1/\epsilon^2} \operatorname{erfc}(1/\epsilon).

Polynomial division of the numerator then gives the closed expression

\boxed{D_{\mathrm{rms}}^2 =1-\frac{\sqrt\pi(1-\epsilon^2/2)}{J(\epsilon)}}.

The boxed formula suffers cancellation if evaluated naively at very small $\epsilon$ . Expanding it instead gives

D_{\mathrm{rms}}= \frac{\sqrt3}{2}\epsilon^2 \left[1-\epsilon^2+3\epsilon^4+O(\epsilon^6)\right].

The $\sqrt3/2$ is what the polynomial division returns, carried down from the exact spectral identity above. In the downloadable calculation, a time-domain Simpson quadrature is checked against the independent frequency-domain integral at 15 values of $\epsilon$ ; the largest RMS discrepancy is $6.48\times10^{-9}$ . Doubling the representative grids changes $D_{\mathrm{rms}}$ by at most $1.3\times10^{-13}$ .

A second parameter answers a different question

The strong-field quantum parameter is

\chi_e=\frac{e\hbar}{m^3c^4} \sqrt{-\left(F_{\mu\nu}p^\nu\right)^2}.

Roughly, it measures the field seen in the electron rest frame relative to the QED critical field. Quantum corrections can become measurable around $\chi_e\sim0.1$ ; $\chi_e\sim1$ is a useful guide for strong recoil. Neither threshold is an abrupt phase transition.

In the normalized prescribed-force model, $\epsilon$ measures the accuracy of classical reduction of order, while $\chi_e$ flags the adequacy of a deterministic classical emission model. Adjust them independently below.

Two-axis diagnosis

“Breakdown” of which approximation?

quantum caution
LL accurate

two warnings

classical safe
LL accurate

classical reduction stressed

epsilon: slow pulse → fast pulsechi-e: classical → quantum

log10 epsilon: -6.0log10 chi-e: -2.0

Diagnosis: This benchmark predicts an accurate classical reduction in a classically safe regime.

The one-percent boundary is a pedagogical threshold, not a universal law. The map keeps two logically independent tests visible; it does not replace a full relativistic or quantum calculation.

For a femtosecond pulse, $\epsilon$ is around $10^{-9}$ , so this benchmark’s LL truncation error is around $10^{-18}$ . A strong-field experiment can still have appreciable quantum corrections because its $\chi_e$ need not be small. Recent all-optical radiation-reaction measurements, for example, report mean quantum parameters around $0.05$ – $0.1$ : already interesting for quantum corrections, but not accurately described as $\chi_e\gtrsim1$ .

What is—and is not—new here

Gaussian-pulse comparisons between AL and LL already exist, and the broader fact that reduction-of-order series may be asymptotic is established. The claim here is narrower: for this normalized Gaussian benchmark we give an exact finite- $\epsilon$ RMS error identity, connect it explicitly to the singular loss of differential order, verify it through independent time- and frequency-domain computations, and place it on a two-parameter diagnostic map without conflating $\epsilon$ with $\chi_e$ .

The model is one-dimensional, nonrelativistic, and externally prescribed. It does not prove a relativistic error theorem, compute a photon spectrum, include beam back-reaction, or resolve the conceptual acausality of the AL physical branch. Values near $\epsilon=1$ are mathematical stress tests, not attainable electron pulse durations.

That limited claim is useful precisely because it is auditable. Every equation needed to reproduce the central result fits on this page; the repository adds the numerical cross-check and complete manuscript source.

Reproduce or challenge it

The public package contains the LaTeX source, figures, generated data, dependency-light Python script, a review note recording the claim boundaries, and a self-contained arXiv upload archive. Run the script and try to break the agreement: enlarge the integration window, refine both grids, or replace the Gaussian and see which parts of the exact derivation survive.