In our blog post on November 10, 2025 we showed that Feynman integrals miscalculate several simple processes in the classical limit.
e- ----------------------------------
| | p - k virtual
| q + k | photons
e+ ----------------------------------
k = arbitrary 4-momentum
in the loop
Let us assume that we have been able to calculate the Feynman integral for the above loopy diagram – possibly renormalizing the result.
Let us upscale the system, making the electron mass me N²-fold and the elementary charge e N-fold, where N is a large positive number. Then 4-momenta scale by the factor N² and the coupling constant e² / (4 π) scales by the factor N².
We consider a small part of the Feynman integral:
d⁴k [photon and electron propagators]
* [coupling constant for 4 vertices].
1. The 4-momentum volume element d⁴k scales by (N²)⁴ = N⁸.
2. The coupling constants e² / (4 π) scale by (N²)⁴ = N⁸.
3. A photon propagator scales by 1 / (N²)² = N⁴.
4. An electron propagator scales by 1 / N².
5. The combined scaling of the propagators is 1 / N⁴ * 1 / N² * 1 / N⁴ * 1 / N² = 1 / N¹².
6. The total scaling of the integral is
N⁸ * N⁸ / N¹² = N⁴.
e- ----------------------------
| q
e+ ----------------------------
The scaling of the simplest tree level scattering diagram is from the coupling constants (N²)² and from the photon propagator 1 / (N²)². That is, the scaling is 1.
The scaling of the loopy diagram is N⁴. The loopy diagram would dominate for large N. This only makes sense if the integral of the loopy diagram is zero.
M. Consoli (1979) calculates box diagram integrals (Figure 7. (a) and (b) in the paper). Consoli does not mention that the integrals would be zero.
We conclude that the classical limit of the Feynman box integral is nonsensical. This suggests that the integral is nonsensical also if we use the normal electron mass me and charge e. What is the problem? Apparently, the Feynman integral is not the right way to analyze the "fine details" of a particle orbit in a Coulomb field. The simplest tree level diagram works very well, but a loop with two virtual photon exchanges is not sensible
the Feynman integral in QED exaggerates the high-energy spectrum of bremsstrahlung, in the classical limit?
m, q
• ----------_____
----->
● M, Q
Let us have a massive, large negative charge q pass a very massive large positive charge Q.
Let the position vector of q be z(t), where t is the time coordinate. Let t be zero when q is closest to Q.
Intuitively, the spectrum of electromagnetic radiation which the negative charge q will emit, depends on the Fourier decomposition of the acceleration vector:
a(t) = d²(z(t)) / dt².
All derivatives of a(t) obviously exist, are continuous and tend to zero as t goes to (negative) infinity. Furthermore, the derivatives behave "nicely" when |t| is small. The theorem in the link states that then the Fourier transform â(k) of a(t) satisfies:
|â(k)| * |k|ⁿ → 0
for all n > 1. That is, high frequencies |k| are suppressed extremely fast, probably exponentially.
But the Feynman formula, which is used to calculate bremsstrahlung, suppresses high |k| quite slowly. It is suppressed by the electron propagator, and the photon propagator in the Feynman diagram (q = e-, Q = Z e+).
We can assume that the particle e- has quite a lot of kinetic energy and passes Ze+ from a relatively large distance. The kinetic energy would allow e- to send very high-frequency bremsstrahlung photons – quantum mechanics does not block that.
The Feynman diagram does not understand how sharp is the turn in the path, caused by q
We wrote about this on November 10, 2025. In the bremsstrahlung diagrams above, the momentum change q can be abrupt or slow, depending on how large Ze is. In the classical limit, the bremsstrahlung is drastically different for large and small values of Z. The Feynman integral does not understand anything about this.
We know from experiments that Feynman integrals do calculate practical applications of bremsstrahlung correctly. The electron e- in them is very far from a classical particle. For a quantum particle, the sharpness of the turn in the electron path does not matter.
Conclusions
Feynman diagrams and integrals miscalculate the classical limit in many basic processes.
On the other hand, for a strictly quantum particles, Feynman diagrams work well – we know that from experiments.
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