Huge problem with classical and quantum electrodynamics

I was thinking to myself today about a variety of ideas not related to my research. One problem which came up is in consideration of a charged particle and the electromagnetic field it generates. I suppose the problem I considered is from a classical sense. Imagine a charged particle such as a proton or electron. Neither particle decays according to theory, and no one has experimentally seen this occur. These particles may exist forever if left alone. There is energy stored in the mass of the particle, of course, but this is certainly finite. The problem I considered is how can a finite energy source give rise to fields which carry energy with them, but do so forever? I think this problem is very similar to some problems outlined in undergraduate and graduate books as well as in many published articles even as recent as 1998 (John David Jackson cites a paper from 1998 which discusses this problem, but the latest edition of his book was published in 1999). 

One problem, which I'm not yet sure deals with the same problem I'm considering, is the problem of a self-force. A radiation force from a moving electron with some decent degree of accuracy, perturbs the trajectory of an electron. The mathematics come in the form of the Abraham-Lorentz formula which describes the radiation reaction force. This approximation only works for certain regimes since it can mathematically have multiple, unrealistic solutions. The force works by considering the fields of the source particle exert a force on the particle while the particle is in motion. Since the fields are generated by the particle, this is in effect like the particle exerting a force on itself. This perturbation term is actually a useful correction despite the physical meaning being awkward and, I think, ridiculous.

Another problem, which I think is related to the self-force problem but is closer to the problem I'm considering, is mentioned in David J. Griffiths undergrad E&M textbook.

  "...the point charges (electrons, say) are given to us ready-made; all we do is move them around. Since we did not put them together, and we cannot take them apart, it is immaterial how much work the process would involve. (Still, the infinite energy of a point charge is a recurring source of embarrassment for electromagnetic theory, afflicting the quantum version as well as the classical. ... Where is the energy, then? Is it stored in the field, ..., or is it stored in the charge...? At the present level, this is simply an unanswerable question: I can tell you what the total energy is, and I can provide you with several different ways to compute it, but it is unnecessary to worry about where the energy is located. In the context of radiation theory (Chapter 11) it is useful (and in General Relativity it is essential) to regard the energy as being stored in the field, ... But in electrostatics one could just as well say it is stored in the charge... The difference is purely a matter of bookkeeping."

To me, his directions to the matter of bookkeeping seem like a cop out. There have been many that have attempted to "fix" the Abraham-Lorentz self-force problem with considerations of relativistic effects, but according to a paper by Rohrlich in 1997, the "pathological" solutions can be made to vanish (but sitll in special regimes). So there is still a problem with the matter of infinite energy, wherever it may be...I think. No one sounds entirely convincing even if you ask the experts.

So, what gives?

Adam D Scott

Center for Neurodynamics
Department of Physics & Astronomy
University of Missouri at St. Louis
http://www.umsl.edu/~neurodyn/students/scott.html

rounding out dissertation plans

So I've been working and playing and studying as usual this semester.  My course in nonlinear dynamics is going swimmingly; partly because I've gone through most of the book we're working out of this semester. :)  My research is gaining steam on both fronts - evolution and neural.  Minecraft has taken over my nightly activities, and on occasion, it's taken over a day or two. :\

Anyway, I've begun working on my dissertation proposal and outlining my first paper for publication.  My dissertation will include three parts, two on the evolution model we have, and one on a neural model previously used in our lab.  

The first part of the evolution model will probably focus on cluster (species) activity on even fitness landscapes (all organisms produce the same number of offspring).  This is important in addressing two questions.  The first deals with the problem of whether species really form when there is no landscape to determine what is most fit for the organisms.  Generally, natural selection is considered to take place when the environment organisms live in, along with their natural ability to survive in said environment, generates a selection criteria for its organisms.  The selection criteria in our model is determined by the gradient, or landscape, which determines how many offspring a nearby organism may have - their fitness.   If you take away the gradient of fitness, then there is effectively no natural selection.  However, the organisms still mutate each generation as dictated by their mutability and generate a diverse set of species.  This orientation of the system should be that of neutral theory, such that diversity arises randomly.  Although the mating algorithm of our model intrinsically produces species, I will probably explore how distinguishable those species are throughout the simulations.  I suspect that under the condition that every organism in the starting population receives a unique mutability value among a wide range of possible values, the species will go through many complex interactions - making them very inconsistent over many generations.  After what might be considered transience, the available organisms will have dwindled their competitors to just a handful of mutabilities.  This should reduce the amount of species interactions, so the species will become much more consistent and distinguishable (few interactions with other species).  (Note that I don't really have sources that discuss species interactions, so this idea will most likely change.)

The second question addressed with this portion is whether there is a "best" mutability even in the case of neutral theory.  I already briefly touched on this idea near the end of the previous paragraph.  My intuition suggests that as the organisms compete, no particular set of mutabilities will survive the full simulation.  My reasoning is that because there is no selection criteria, there should be no "best" mutability, as long as the organisms can avoid an imposed overpopulation density condition (kill off those too close to an organism).  I think my data is already showing that there is a bimodal distribution of surviving mutabilties, which implies that there are two mutabilities which are optimal for survival.  The tricky thing is that in all landscape situations, I keep seeing a bimodal distribution of survived mutabilities approximately the same in all cases.  I don't yet have a reason why this could be happening.

For the species interaction, I'll probably look at it under the scope of no competition when all organisms are given the same mutability.  I have a simple prediction for this.  As mutability increases, species interactions will be very rare at first, but then, with a large enough mutability provided, the species will move to a state where they nearly always interact with other species.  Hopefully, there should be a small range of mutabilities which will indicate this change in such a way that it can be modeled as a phase transition (like solid to liquid to gas sort of idea).  Perhaps I'll even come up with a sort of kinetic energy analogy for mutability and use the fitness landscape to define a potential energy so that I can use statistical physics and thermodynamics principles to model it.

For the neural project, I'll attempt to model glutamate activity such that neurons in a network desynchronize as the synaptic activity of glutamate falls all while in conditions prone to epileptic activity (neurons synchronize).  The term in the model which determines coupling between neurons is that of the synapse.  From this term, the strength of influence connected neurons have on each other gives rise to the possibility of synchronization.  Luckily, a previous grad student in the lab showed how to gain synchronization in such a way, that it corresponds well with experiments we've done on rats.  The change imposed to synchronize the network is the same which models seizure activity (reduced potassium conductance).  In the experiments, I noticed that there seem to be several characteristic orchestrations of the local field potential.  The behaviors can be very different, particularly with the endings.  The activity may cease abruptly or gradually decay.  Furthermore, the length of seizures may vary from tens of seconds to several minutes.  These behaviors led me to wonder about HOW the network synchronizes and desynchronizes.  There is some literature to back up more specifically the idea I considered more abstractly, which is that glutamate variation changes the coupling strength between neurons.  There are a few things to consider with this.  First, when excitatory neurons fire, they tend to release glutamate, which is an excitatory neurotransmitter.  This causes the post-synaptic neurons to increase their potential to fire an action potential as well, thus influencing when connected neurons may fire (driving mechanism to synchrony).  However, each neuron has a limited supply of glutamate in vesicles to release, and require supplementation of new glutamate to release and package for further synaptic activity.  These processes take time, and may be over long enough periods that allow the network to become effectively disconnected enough so that synchrony of the neurons is lost.  Currently, the Wilson model which I will use fixes synaptic conductance (coupling strength).  It will be my job to determine an effective model for how synaptic conductance varies in time.  Hopefully, the parameters needed are physically plausible within the conditions imposed to garner seizure activity.

Perhaps another detail I can include in my model is that the potassium conductance reduction need only be applied to a small localized subset of the network, thus modeling the experimental system more closely.

SO, yet another long winded explanation of thoughts, but I think I can turn this into the basis for my dissertation proposal.  Assuming I really do that, then I can say this mission is a success!

fixing ideas on dark matter and other cosmos stuff

These question and answers are from between Dr. Cheng and myself. Of
course, I'm asking questions and he's answering after the ***. I had
the complete wrong idea about dark matter....oops.

First, if dark matter is attributed to the expansion of the universe
and dark energy is attributed with an accelerating universe via the
cosmological constant, then how are they not directly related?
***Dark matter, just like ordinary matter, is subject to gravitational
attraction, while dark energy, to gravitational REPULSION. In our
universe there are (4%) ordinary matter/energy (called baryonic matter),
(21%) dark matter and (75%) dark energy. So dark matter and dark energy
are NOT, under our present understanding, "directly related".

Second, what are the chances that the expansion is not due to vacuum
energy? Kari said that some have tried pinning expansion to vacuum
energy, but that the vacuum energy is orders of magnitude less than
what is needed to achieve what we observe. However, would this vacuum
energy need to be handled as nonuniform if one considers that space is
warped? Whoever looked into this, how did they handle consideration
of vacuum energy? Does it even change according to the warped-ness of
space? (pardon the layered questions)
*** Cosmological constant is the name of the math term in Einstein's
equation that has the effect of being gravitational repulsive. Its most
probable PHYSICAL interpretation: "it's the energy of the vacuum". But a
straightforward calculation shows that the quantum mechanical vacuum
energy is 120 orders of magnitude too large compared to the observed
amount of dark energy (NOT too small). If it is the cosmological
constant, the warped-ness of spacetime will not bring any nonuniformity
in dark energy.

Third, I read somewhere that there are drag effects of objects
orbiting in space. Could a more fluid-like consideration of space
give rise to the expanding universe (high/low pressure systems
depending on empty/filled space or rotating vortices)?
*** Yes a rotating gravitational source can drag the spacetime around
it. But all this is consistently accounted for in the context of general
relativistic description of the expanding universe.

Lastly, is dark matter thought to exist as a constant amount; if not,
where might it come from?
*** Dark matter is definitely not uniform. In fact the present
understanding of the observed cosmological structure (galaxies, clusters
of galaxies, voids...) is built on the idea that structure formation
started among the dark matter first (from gravitational clumping), then
the baryonic matter falls into the grav. Potential wells formed by dark
matter. The favored idea of the origin is that they are the cosmological
thermal relics (just like the cosmological microwave background
radiation).

slacker

So, I've been thinking a lot lately about various things.
1. How exactly the universe expands.
2. How gravitation works.
3. 4-dimensional cross-product.
4. How to experimentally make sense of critical slowing down.
5. Ion channel desensitization vs. sensory adaptation.
6. Developing apps.
7. Wondering why I don't follow through with any of these ideas...or
at least why I take so long in addressing them.

I'll go through each topic, but perhaps I won't do it all in this
post...that could make for a very long post.

1. I think the current explanation among many astronomers and
cosmologists is that the universe expands because of dark energy and
matter. Dark matter is like gravity, but it works in reverse. What I
don't understand is where this stuff comes from. Can it's effects be
attributed to something else? Take this site's explanation, for
example: http://www.physlink.com/education/askexperts/ae404.cfm
Although, "where" the stuff is has been mapped according to
measurements of a galactic supercluster by Hubble:
http://hubblesite.org/newscenter/archive/releases/2007/01/image/a/
I don't think it yet answers my questions though. I certainly don't
think it proves anything...whether dark matter is real or not. It
seems to strongly suggest that it's out there though. It may still be
just a coincidence of currently known forces; although explaining the
observed effects with what we have in our toolbox doesn't seem to work
well enough. Perhaps that just means we need to expand what we know
instead of creating something new. I don't think it is impossible for
us to create a new tool to explain the cosmos, but in the end realize
it is just a special case of tools we already have.
My loosely assembled, guesswork hypothesis is that if the amount of
dark energy in our universe is not fixed, then it must come from
somewhere. That somewhere might be a higher dimension, but then
anything existing in that higher dimension must be losing that energy.
This would imply a conservation of energy among all dimensions. This
would allow the dark stuff to infiltrate our known dimensions and
possibly give rise to the cosmological constant (but this would only
be to accelerate the universe...whatever that really means). If the
amount of dark energy is actually fixed, then perhaps it is diffusing
as suggested by the recent Hubble map and subsequent measurements may
indicate. BUT if dark energy is really an occurrence of stuff we
already can measure and know about, then one of two things may happen
(although I'm not sure if they'd be mutually exclusive). Either it
comes about from energy associated with vacuum or it comes about
because of gravitational effects on space itself...like fluid effects.

I'll have to finish talking about this later. Have to go to UMSL.

Edit to finish:

I talked with an astronomer student in the lab next to mine, and I asked her about dark matter and energy. I need to be careful with the two terms since they are not linked like typical matter is with energy we deal with everyday. "Light" matter can be related to energy by it's mass with everyone's favorite E=mc^2, where E is the energy of a mass of matter, m, and c is the speed of light in vacuum. Dark matter and energy don't have this sort of relation...direct correlation...as far as anyone knows. Dark matter does give rise to the expansion of the universe as I said above, but dark energy is strictly associated with the cosmological constant that possibly accelerates the universe.................I have trouble with this still. They seem to describe the same action, but apparently there's a big difference that I'm not getting.

I emailed some questions about this stuff to a professor in my department...supposedly he'd know best when it comes to this stuff. Dr. Ta-Pei Cheng, I'm counting on you!

quantum entanglement and information transfer

I have been reading about the EPR paradox (Einstein, Podolsky and Rosen) and the implications of quantum entanglement. You can think of entanglement like the drawing of one of two cards. The two cards are entangled in the sense that if I draw a red card while we know the other is blue. Then by checking the card I have drawn, you will know exactly what card is left, the blue card. This concept is the same when talking about two objects in entangled quantum states. If I am on the moon with a particle which may be in state 1 or 2, and you are on the earth with a different particle in state 3 or 4, but our pair of particles may only be in an exclusive combo state of either 1 and 3 or 2 and 4, then if you measure the state of your particle, I need only to ask you what your result is to determine the state of mine. HOWEVER, information can only travel at most the speed of light. SO, if we measure our particles simultaneously, then we should be able to have the possibility of obtaining a result which does not match either of the two possible states I listed before (1 and 3 or 2 and 4), since information of your particle and my particle will not be able to reach the other in time to "let the other particle know" that something has changed or that a measurement has occurred. I am not convinced that entanglement can exist at such distances. Furthermore, how does the effect of measurement influence entanglement (by measuring particles or systems, we effectively put our system into a certain state...from which it may then evolve according to the state we measure it in).

critical states

Do we live in a perpetual or constant world that is in the critical state? Is there any difference between "action at a distance" and microscopic forces influencing macroscopic properties and behavior?

If I have a ferromagnetic material near its Curie temperature and change one electron's spin direction, then it should have an effect on another electron's spin at any range from it. Certainly, the force that the change in spin of the electron I choose first influences to some degree its neighboring electrons, then they influence their neighbors and so on until they find electrons at any distance to influence a complete change in direction. The influential role of the electron I have chosen is merely a fluke of probabilities in my eyes while certain in the eyes of nature.

However, is this any different than if I were to blow a feather off a table. My lungs create a pressure change which causes a chain reaction of colliding air molecules which in the end, and along the direction from my mouth to the feather, is just events that can be described microscopically in order to affect the feather from a distance. To provide another example of microscopic local influences causing global reactions, consider social networks.

My undergraduate research advisor, Dr. Ojakangas, would tell his students about some physical law or theory, then asked, "Do you buy that? Because that's all I'm selling!" Now, when I tell some story or give a lecture, I might ask whomever I am talking to, "You buy that? That's what I'm selling!"...or something to that effect. The point is this: after Ojakangas fed us (his Mechanics II students) that line the first time, he told us that he had heard a professor of his say that (I think at CalTech). He apparently liked it, so he made a similar comment to us. I like it as well, so I make the comment to whomever cares to listen to me on occasion. I imagine that by this point, others either those who have heard Ojakangas' professor at CalTech, Ojakangas, or myself say this line have or will also say this to others. Other people who have no idea where the source of the silly line came from. This is like action at a distance. (I'm not going to claim that the source is even Ojakangas' professor; that's just as far as I know who came up with what!)

An even bigger social network analogy is that of the internet. Before the internet one person with a video of something ridiculous would only be able to show the people they knew and not too many others. Now, that video can go viral and effect millions of people who have absolutely no direct connection to that person. The internet has provided a way to make a correlation distance between people in the world near its maximum possible value, just as in the critical state of electrons in a ferromagnetic material near the Curie temperature.

Does this all mean that we live in a constant state of criticality? Where the butterfly effect really changes everything? If not everything, does it at least make great dents in the previous order that existed? Has there ever been order? I suppose when talking about correlation distances of one object or idea influencing another at a distance, we must consider correlation times. For our brief time on this earth, most of us probably won't cause the global changes in our lifetimes, but our actions now may influence the next generations in ways we would not expect. Stories your parents may have told you about their times in life may influence the way you conduct yourself. Your actions may then influence others which may lead to global implications later, like the leaders of nations deciding between good and evil.

My bet is that we live in a constant state of criticality. I think that every action now influences the current order to be reordered. Whether there is an end to the criticality, I doubt it exists. I cannot think of anything on any scale in which a scale's microscopic forces do not propagate to influence objects at a distance. However, the time scale in which to consider universal objects may need to be characteristically near infinite. Keep in mind that pockets of objects that do not seem to be influenced are part of the property which determines criticality. Nothing is globally special, no matter the amount of detail you consider.

species & fractals, linear v nonlinear

So I've been thinking the past few days about speciation and the splitting that occurs. I wonder if phylogenetically, speciation can be fractal under certain conditions.

A model which I'm imagining could at a very concrete and highly unrealistic state begin with a species splitting into three branches. Each of these three branches would then split into three branches, and so on. I doubt it would be too difficult to show or at least understand that this should inevitably lead to a fractal.

In order to arrange some realistic aspect to the model, I would then allow some or all of the three branches to not form. These instances would model extinction of species in a way. Instead of forming, then dying out, the branch just doesn't form. The rate at which this occurs would depend on how many branches are currently able to form (once a branch has divided or failed to branch at all, it would no longer be counted), then a random number of branches that would form will not. I would think this sort of model would retain fractal behavior, but certainly will not look as "nice" as without extinctions.

Another way to make the model more realistic is to allow a variable number of branches, say between 0 and 5 or so, or whatever current estimates on speciation might suggest. This part may destroy the fractal like behavior, but as I have seen in Barnsley's fern, this variable extension of the phylogeny may be fine.

I'm sure there are plenty of other ideas which may be implemented, but I think what I have listed should be reasonable for a simple model. The real question would then be, how closely does an averaging of many simulations of the model reflect the phylogeny of today's species within all higher levels of taxonomy?

Another thing I've been thinking about is linear versus nonlinear systems. Primarily, I've been thinking about why some nonlinear systems cannot be linearized, especially globally. Locally, around fixed points, nonlinear systems may be linearized by evaluating the Jacobian matrix at the fixed points. However, what restricts us from taking the nonlinear terms and calling them a new term by a change of variables? Certainly, most, if not all, cases will result in a system with more dimensions (each independent variable gets its own dimension). However, if nonlinear terms are linearly independent among the linear terms, then could a new system be generated in order to make the system easier to solve? Perhaps this is all bogus because the nonlinear terms may be formed by the linear terms, which would then be a case of ALL nonlinear terms being linearly dependent which would then disallow one to make a change of variables of the nonlinear terms in order to present a new dimension to the problem.

Another idea related to this is to explore what is needed in order to execute linearization of a system- as in constructing the Jacobian matrix evaluated at the fixed points of a system. Following along with the steps indicated by Strogatz in his book, Nonlinear Dynamics and Chaos, I found (and am assuming) that the only requirements needed for a two dimensional system with arbitrary coupling are that the transformation functions used in the change of variables need to be differentiable and have an inverse which is also differentiable.

For example, let the derivatives of x and y be x' = f(x,y) and y' = g(x,y), and let the change of variables, u and v, be u = F(x,x*) and v = G(y,y*), where x* and y* are fixed points. Rewriting functions for x and y gives: x = Finv(u,x*) and y = Ginv(v,y*), where Finv and Ginv are the inverse functions for F and G. Differentiating u and v gives: u' = F' + u' F and v' = G' + v' G. Rearranging for Finv and Ginv, then differentiating gives: x' = Finv' + u' Finv = Finv' + u' x and y' = Ginv' + v' Ginv = Ginv' + v' y. By substitution of these new x' and y' equations with the originals gives: f = Finv' + u' x and g = Ginv' + v' y. Solving these for u' and v' and then expanding them as Taylor series evaluated at the fixed points should give appropriate Jacobian matrices from which an analysis of the eigen values will determine what sort of behavior can be expected.