∫ 1 / k⁴ dV.
k ∈ ℝ⁴
The paper in the fnal.gov link uses dimensional regularization in 4 - 2 ε dimensions. At the end of the calculation, the paper states that diverging terms cancel each other.
The paper in the utexas.edu link utilizes various symmetries, and states at the end of the calculation that the integral converges.
We conclude that the integral probably is benign: if the integral is summed in the "natural" order of increasing 4-momenta in the virtual photon, then the integral will converge. There is no need for regularization or renormalization.
In the previous blog post we claimed that the mass-energy of the electric field of the electron is
α / (2 π) * me ≈ 1/861 me.
Since the integral is benign, we can claim that the result is robust: the result does not depend on dubious regularization or renormalization procedures.
The quantum imitation principle
Quantum imitation principle. Quantum mechanics tries to imitate classical mechanics. The resolution of the imitation is restricted by the Compton wavelength associated with the energy available in the process. The imitation may in some cases be more accurate, if there is a lucky coincidence. The imitation is further restricted by quantization.
We introduced the principle above in our previous blog post. Quantum mechanics tries to imitate the energy of the electric field of the electron. But the resolution is quite poor: the Compton wavelength of the electron is
2.4 * 10⁻¹² m,
which is a large value compared to the classical radius of the electron 2.8 * 10⁻¹⁵ m.
The imitation of the electron electric field only succeeds at the resolution of the Compton wavelength. Quantum mechanics believes that the mass-energy of the electric field is just 1/861 of the electron mass.
The fact that we do not need regularization or renormalization in the calculation of the Feynman integral, stresses that the quantum imitation principle is robust: it will produce finite values without dubious mathematical procedures.
What if descructive interference cancels diverging integrals?
For classical waves, a wave whose frequency is
f
cannot normally produce significant waves whose frequency is > f. If we use Green's functions to construct a solution, destructive interference, in a typical case, cancels all high frequencies.
The cancellation is not absolute: if we would sum the absolute values of each wave, the integral would diverge.
We conclude that it is ok if the integral diverges in absolute values, as long as it converges when integrated in the natural order of increasing 4-momenta.
Let us calculate an example. The impulse response, i.e., the Green's function for a static point charge is
∫ 1 / k² * Real( exp(i k • r) ) dV,
k ∈ ℝ³
where dV denotes a volume element of ℝ³, and Real takes the real part. The integral k³ / k² diverges badly. If we use Green's functions, we cannot expect absolute convergence of integrals.
If classical wave processes cancel high-frequency waves, why do high-frequency waves remain in Feynman integrals and cause divergences?
In this blog we have long suspected that divergences are a result from a wrong way to apply Green's functions to scattering phenomena.
The reason may be that we calculate Feynman diagrams in the "momentum space", and ignore the position of the particles.
In the case of vacuum polarization, we claimed in 2021 that the divergence comes from a sign error when considering the Dirac hole theory.
*** WORK IN PROGRESS ***
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