I really like classical mechanics. It's neat. My thoughts on the matter are best summed up by the following quote:
Vladamir Alexandrovitch Fock
A professor explained to me that a distribution of electrons on a sphere will spread out to maximize their distances. I asked her if one could solve for the trajectories of the electrons which give rise to this phenomenon explicitly, because I had hoped to see it happen "in-real-time". I was a fool, an absolute child. Not really, but it turns out that it is actually very difficult to present a closed form solution for the trajectories of charged particles coupled to fields.
It came as great shock to me when I learned that the joint-evolution problem for a point charge and its field is not even well-posed, with the reason behind the ill-posedness mathematically simple enough that one could explain it to a high schooler. When I learned that we spent hundreds of years studying equations about the motion of particles and fields without being able to actually solve for their trajectories, I looked the other way. But now you're telling me that those trajectories aren't even well-defined? Like, they don't even exist? Why the heck have we been so tirelessly studying this ill-posed theory of physics for the last 200 years?
Despite the fact that the fundamental axioms of the theory are not consistent (for point charges), it has still provided us with fantastic results time and time again, and allowed humans to progress to a technological standard that would be unimaginable at the theory's time of discovery.
Todd Howard
Honestly, I've never really cared about advancing society. Like, it's cool but not my jazz. I think one of the main reasons why I really disliked electromagnetism during my undergrad was this focus on "applications to the real world". Newtonian mechanics was presented to me as a theory that was worth studying because it was beautiful on its own, regardless of applicability to real life. But electromagnetism was the other way around; everything about it had to be motivated by real life because unlike Newtonian mechanics, "this one is actually right this time". When I learned that Maxwell-Lorentz isn't actually a mathematically coherent theory, I felt simultaneously vindicated and disapointed.
James Clerk Maxwell
The problem with the Maxwell-Lorentz theory of electromagnetism lies within the infinite self-force that a charged particle will exert on itself. A charge generates an electric field whose magnitude is proportional to the inverse square distance from the particle, and the Lorentz force acting on the particle is given by the electric field evaluated at the charge's position multiplied by the charge's value. This force is not defined because the electric field is infinite at the charge's position. In classrooms we are taught to ignore the infinite term, because "the particle can't exert a force on itself". Many don't question this. But that's not how the theory is supposed to work. The particle isn't exerting a force on itself, rather it is the electric field which is exerting the force. The particle also may be sourcing a term in that electric field, but this does not change the definition of our force law. To ignore the self-force term, one must break the principle of super-position.
Detlef Durr and Sergio Albeverio, late 1980s
Hans Reichenbach
Michael Kiessling performed a rigorous analysis of the self-forces which a charged particle will undergo in various theories of electromagnetism, and presented the unique force law which preserves the total energy of the particle-field system. Hoang et al. showed that for BLTP, the joint evolution problem for a charged particle and its fields is well-posed, meaning there exists a unique solution to the dynamical problem. The self-force which a particle exerts on itself in BLTP was found to be depend on the velocity of the particle, in such a way that the force is zero if the particle is at rest in vacuum.
In my page on Mathematical Physics, I claim that there are three steps in the process of physics research. The third step is to analyze the mathematical results, and translate it back into the language of physics.
The physical explanation behind these self-forces is as given. A charged particle whose motion is perturbed (perhaps by some background radiation) will also generate motion in the field it is sourcing. This generated field motion carries energy of its own, so without a self-force one would find that the total energy of the system would not be conserved. The self-force is the corrective term which ensures that the total energy of the system is conserved. If the particle generates "field energy" due to its motion, then the self-force must generate equal and opposite "particle energy" to compensate. This is why it is called "Back Reaction". The self-forces which were found by Kiessling and proven well posed by Hoang et al. are not only physical, but are necessary for any physical theory of fields sourced by point particles.
John Herschel
My research has been studying a toy model 1 space dimensional version of the Joint Evolution problem for a charged particle. There are two reasons why we chose to conduct this research. First of all, everything is hard in three dimensions. Secondly, the Maxwell-Lorentz theory of electromagnetism in 1 space dimension is a lot less singular than its 3-D counterpart, so we hoped that it would lend itself to an easier and less singular self-force analysis. We were correct. Shadi Tahvildar-Zadeh, Samuel Leigh, and I were able to successfully write down a very nice and closed form for the self-force which a charged 1-d particle "exerts on itself", and proved well-posedness for the Joint Evolution problem.
To see why charged particles experience self-forces, consider a stationary charged particle whose motion is purturbed. The motion of the charge's position will generate motion in the field it sources, and this field motion will carry energy. To conserve the total energy of the system, the field's kinetic energy must be drawn from the particle's kinetic energy, and this phenomenon effectively gives rise to a self-force that is restorative. We show that in the case of a scalar field in 1 space dimension, the self-force is restorative, proportional to the velocity of the particle, and that the particle tends rest asymptotically.
The pictures above show the trajectory of a particle which is being temporarily accelerated by external radiation, and then returning to rest due its self force. The self force is cool, but its presence is a bit exaggerated here because our particle is moving very close to the speed of light.
Norbert Wiener
There are currently 100 different routes we can take from here. We could try generalizing to the multi-particle case, however that's going to involve studying delay differential equations. We can analyze the dynamics of the particle and see if there's a hamiltonian formulation which includes the velocity dependent self-force, and then quantize this. But that would require quantizing terms for which is not clear how to do so, such as position times momentum.
Lastly, we can do what we originally intended to do with this project, which was to attempt to study this system in the context of 1 dimensional general relativity. The problem is that even though 1-D G.R is a lot less singular than 3-D, we still found it to be too singular for it to give rise to a well-posed dynamical problem. Once again, it is because the curvature of spacetime which the particle generates around itself causes the dynamical equations to become undefined. We may look into alternative BLTP type versions of 1-D G.R to resolve this problem. Alternatively, we may instead opt to study the problem of a 1-D electron orbiting around a 1-D fixed proton. The proton will generate a fixed background space-time while also sourcing an electromagnetic field which acts on the electron in addition to the electrons own self-forces. If we show that this system is well-posed, we may be able to then quantize this, and finally accomplish our dream of studying a quantum mechanical system on curved space-time.
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Gif of moving charged particle