# PhD Work

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# Electron self-energy paradox

The **electron self-energy paradox** says that the energy required to build an electron is infinite.

`Energy(electron) \rightarrow \infty`

The paradox emerges from classical electromagnetism because the math is too niave to capture the complexities of fundamental particles. One way to think about it to ask where the charge resides in a fundamental particle. Because, our classical way of thinking of electromagnetism is to reduce the object into smaller, more fundamental particles. But there is nothing more fundamental than a fundamental particle. So, our mathetmatical techniques fail.

The Nobel Prize in Physics 1965 was awarded jointly to Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.” However, this what is the necessity to pursue a different model of the electron when the point charge has led to great success with QED? To answer, I quote Feynman summarizing QED.

“It seems that very little physical intuition has yet been developed I this subject. In nearly every case we are reduced to computing exactly the coefficient of some specific term. We have no way to get a general idea of the result to be expected. To make my view clearer, consider, for example, the anomalous electron moment…We have no physical picture by which we can easily see that the correction is roughly α/2π; in fact, we do not even know why the sign is positive (other than by computing it)…We have been computing terms like a blind man exploring a new room, but soon we must develop some concept of this room as a whole, and to have some general idea of what is contained in it. As a specific challenge, is there any method of computer the anomalous moment of the electron which, on first rough approximation, gives a fair approximation to the α term?”

QED, even with its great success, never addressed the paradox of the point charge—it only circumvented it through mass renormalization. The infinities that emerge in QED also emerge in classical electrodynamics. Therefore, the necessity to pursue another model of the electron stems from the lack of classical understanding. Our goal is to seek a deeper understanding of the electron and the corrections provided by QED by developing a classical renormalization procedure.

## A Classical Rendition of the Polarizable Vacuum

The electron self-energy paradox states that the energy required to build an electron is infinite. A resolution to the self-energy paradox can be achieved by re-deriving the electron’s electric potential while assuming the vacuum of space to be filled with an unknown, polarizable, dielectric material. In the standard model, this effective medium can be viewed as being comprised of particle-antiparticle pairs (dipoles) that are emergent from the vacuum. Here, we demonstrate how the vacuum polarizability can be used to derive a “quantum” correction to the classical Coulomb potential. Our result asymptotically reduces to the Coulomb potential for distances larger than the classical electron radius.

(a)Total charge density according to the renormalized solution through the notion of a polarizable vacuum. (b) The new effective potential `\Phi_{VP}/\Phi_0 = 1-exp(-a/2r)`

is compared to the electrostatic Coloumb potential `\Phi_C = q/4\pi\varepsilon_0 r`

.

Applying this general solution to the electron yields

(a)Total charge density according to the renormalized solution through the notion of a polarizable vacuum. (b) The new effective potential `\Phi_{VP}/\Phi_0 = 1-exp(-a/2r)`

is compared to the electrostatic Coloumb potential `\Phi_C = q/4\pi\varepsilon_0 r`

.

## Classical Renormalization of the Point Charge

We extend the classical rendition of the polarizable vacuum to positronium. Our model of positronium shows that first order corrections are related to the so-called Darwin term in relativistic quantum mechanics. This is the first classical theorem that can successfully account for pair annihilation, as is commonly encountered with PET imaging.

Plotted is the interaction potential $V_int$. At large distances the interaction potential converges to the Coulomb potential. (b) The total energy is plotted as a function of separation distance. As the two particles approach (`R/a \rightarrow 0`

) each other and the separation goes to zero, the total energy goes to zero at the origin. This is synonymous with annihilation. For the first time, we have a self-consistent description of fundamental particle interactions without infinities and including pair annihilation.

## Future Work

The simpler case of two interacting particles compared to the interactions of five particles. IMAGE SOURCE

The N-Body Problem doesn’t currently have an analytical solution and is computationally expensive.