We want to find out which phenomena in QED depend on the Planck constant h, and which are classical in nature.
Stanley J. Brodsky and Paul Hoyer (2011) study which calculations with bound states are affected if we let the Planck constant h approach zero. The result depends on which parameters we take to depend on h.
Tree diagrams are generally agreed to be classical. Feynman formulas for them do not depend on h. Robert C. Helling writes about their use in classical field equations.
The simplest Coulomb scattering diagram
momentum p momentum p'
e- -----------------------------------------------
| virtual
| photon
| q
Z+ ----------------------------------------------
Rather than letting h approach zero, we can increase the mass of the electron by a factor 2 and its charge by a factor sqrt(2). The probability distribution of the deflection angle of the electron should stay the same if the process is classical.
Andrzej Pokraka has calculated Coulomb scattering for us. In his formula (3), the matrix element M is the scattered wave per solid angle. The value of
u-bar(p') γ^μ u(p) ~ m_e,
and
A-tilde_μ(p' - p) ~ e / m_e².
Thus,
M ~ e² m_e / m_e².
We see that M stays constant if we double m_e and multiply e by sqrt(2).
The electric vertex correction
k
~~~~~~~
p / \ p'
e- ---------------------------------------
| virtual
| photon
| q
Z+ --------------------------------------
In the vertex correction we have three extra internal lines, as well as two extra vertices, compared to the simplest diagram.
From the Internet we find several calculations of the electric form factor F₁(q²). The calculations are very complicated. Furthermore, regularization and renormalization is used. Renormalization is applied to make F₁(0) equal to 1.
We do not know if the final results of these calculations have been tested empirically.
The final results do not stay constant if we double the mass m_e of the electron and multiply the elementary charge e by sqrt(2). That would mean that the final results break the classical limit. Is the physical model in the calculations wrong?
The Sudakov double logarithm
The leading terms in the electric form factor for large q are
F₁(q²) = 1 - α / (4 π) ( log(-q² / m_e²) )²,
where the log squared is known as the Sudakov double logarithm.
A very large q requires that the electron has very large initial momentum, and passes extremely close to another charge. It might be a situation where the electron is detached from its static electric field. According to our previous blog post, the mass of the electron may appear as 1.022 MeV. If the kinetic energy is much larger than 1.022 MeV, the change in the electron mass should have a very small effect.
We have to check what empirical measurements say about such high-energy collisions.
Sudakov double logarithms seem to cancel in many cases.
Conclusions
In this blog we have the hypothesis that Feynman diagrams are mostly a classical thing. Calculations for the electric vertex function seem to break this hypothesis. The calculations also seem to break the classical limit where we increase the electron mass and charge.
In previous blog posts we asked why a Feynman diagram (with regularization) calculates the vertex correction right. It might be that it does not calculate it right.
We will look at vacuum polarization loops in a subsequent blog post.
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