In short, even if I want to be a mathematical physicist, with the emphasis on the more abstract constructs (fiber bundles, differential geometry, etc.) I need to be firmly grounded in “physics on the ground”, or I’m not likely to have anything useful to say to the physicists.

Plus, from my brief interactions with the professor of the course, I am pretty sure this is the sort of course that will appeal to me.  He has been described as very rigorous and apparently in the HW problems you have to give justifications for each step of the solution.  That is exactly what Ive realized I need to do with just about every problem I work through myself.  Not just rearranging symbols, which you can trick yourself into thinking you understand, but giving clarifying statements and explanations for each step.  What does this theorem really mean?  Why is it a reasonable thing to expand this function in terms of a Taylor Series to solve the problem?

During the fall semester of my senior year at the University of Maryland, I made the decision to forgo graduate school for a year or two and apply to Teach-For-America, where I would (in my own mind) help lift young school children out of poverty by my sheer enthusiasm for physics and math.  Unfortunately for me, and perhaps fortunately for those children, Teach-For-America did not believe I would be able to accomplish such an ideological feat.  And so it was that I found myself at the end of my senior year without a plan for the coming year.

What began as a year or two hiatus from physics turned into three, then four, five…now almost ten years later I have moved around the country several times, studied stratospheric and tropospheric ozone, spent a month hiking in Alaska, gotten married, bought a house, passed the patent bar exam and written over a hundred patents.  And all the while I have felt this nagging feeling that I’m not done with physics and it isn’t done with me.  I have spent some of my free time trying to teach myself some physics and mathematics.  But the results have been mixed.  It’s taken me several years to realize that desire is not enough.  I am lacking many crucial ingredients to learning and doing research in physics:  a strong community of other people who love and are devoted to physics, the structured progression of courses and the direction of professors, advisors and colleagues.

I would like the opportunity now to continue that desire to do research in physics that began when I was an undergraduate.  During my time at Maryland, I sought out opportunities to do research and study under faculty members.  I spent my junior and senior years working with Dr. [ ] Chang in the experimental Nuclear Physics Group.  There I picked up some FORTRAN and helped Dr. Chang run various numerical simulations for [ ] experiment.  I also spent a summer doing some independent study under the supervision of Dr. Xianijong Ji, in the Theoretical Quarks Hadrons and Nuclei Group at Maryland.  On one occasion that summer, Dr. Ji asked me to factor out an expansion of an S-Matrix element he had given me in to a particular form.  I sat with the problem for several days, all the while Dr. Ji kept asking me if I had finished the problem.  When I showed him my result, he explained that the answer was much simpler than the form I had come up with.  So I continued working on the problem, trying to understand my mistake.  Finally, after several days, I went to his office and stated emphatically that the answer I’d worked out was the correct one.  He argued with me about if for several minutes, until I explicitly showed him why the solution was correct.  ”Well, then, congratulations on solving your first non-trivial problem,” he said, which I have never forgotten.  That problem wasn’t a particularly complicated one, but I believe that situation illustrates well my commitment to a problem and my willingness to challenge [?].

After finishing my degree at Maryland, I took the first job that came along.  The position was as a Faculty Research Assistant at the University of Maryland, where I worked under the supervision of Dr. ___ in Maryland’s ____ and Dr. Anne Thompson at Goddard.  During my time as a Research Assistant, I worked on modifying a radiative transfer code to quantify atmospheric ozone concentrations from satellite measurements.  As a Research Assistant I learned IDL, unix scripts, and how to work with very large datasets.  I also gained a lot of experience relevant to the logistics of doing research.  I helped Dr. Anne Thompson in submitting research summaries and grant proposals.  I also co-edited a special edition journal [?], which required me to edit manuscripts, as well as set up and maintain author and reviewer correspondences.

Three years after graduating, Dr. Anne Thompson took a position as a full professor at Penn State, and invited me to come do a masters thesis there under her supervision.  As my soon to be wife had just started her graduate work at Penn State the year before, this opportunity was a no-brainer.  So I spent two years at Penn State studying tropospheric ozone under Dr. Thompson and also married my wife during that time.  For the two years that I spent as a graduate research assistant, I made every effort to come up with new and interesting solutions to the research problems being explored by our group.  A big problem in atmospheric science is the ovewhelming amount of data that is provided by satellite measurements as well as surface instruments.  Sorting through all the data to find patterns can be difficult.  In studying troposheric ozone, we had access to a large collection of ozonesondes, which are profiles of ozone content (as well as other parameters such as water content, temperature, etc.) that are collected during the ascension of a balloon from the surface into the stratosphere.  To help us with identifying possibly interesting patterns and relationships among these various ozonesondes, I developed a clustering approach using MatLab that was adapted from a paper published by another ozonesonde researcher in the field.  I spent several months trying to fine tune the clustering technique so that it would sort the ozonesondes into stable groups in order to find possible underlying relationships between them.  Although the end result was not as robust as I had hoped, I felt I learned a great deal about research during that period and ultimately helped to provide support for several ideas about tropospheric ozone being considered by my advisor.

It was also during this time that my first (and only) research paper was published.  The paper, entitled “Intercontinental Chemical Transport Experiment Ozonesonde Network Study (IONS) 2004: 1. Summertime upper troposphere/lower stratosphere ozone over northeastern North America”, which was based on my masters thesis work, was published in the Journal of Geophysical Review in 2007.

Although I found atmospheric chemistry interesting, I was not nearly as passionate about the subject and research as I had been about physics during my undergraduate years.  I made the decision that upon graduation I would look for other opportunities for employment.  Immediately after graduating with my masters I was presented with the opportunity to do work with a small law firm in Bethesda, Maryland.  Because my wife was still finishing her PhD at Penn State, this job was ideal since it allowed me the flexibility of working from anywhere in the country.

For the last 3 1/2 years I have worked as a patent writer.  In ___, I passed the patent bar exam and officially became a patent agent.  My primary responsibilities have been to write patents for our clients as well as to provide general counseling regarding issues of intellectual property.  It would be difficult to argue that my job as a patent agent is directly related to research in physics.  However, there are several ways in which I believe that my work as a patent agent has connections to teaching, which I feel is also an important part of academic life and which I would like to do as a professor one day.   As a patent writer for the last several years, my primary responsibility has been to explain (using words and figures) complex technical inventions in a way that can be comprehended by someone without expertise in the technological field of invention.  Furthermore, I have a responsibility to boil down the invention to its essence – some generalized features that can be modified in a myriad of ways without changing the original intentions of the inventor.  I believe that these are two qualities that a good teacher should possess: to break down a complicated subject into more easily understandable pieces and to highlight those essential features, the motifs, that provide a backbone for more sophisticated topics down the road.  I am excited about the prospect of teaching other students not just the formulas and laws, but the way of thinking about problems that is common to much of physics.

I am primarily interested in doing theoretical work in physics.  I have great admiration for experimentalists, but I have always felt the most fulfillment in solving a problem using mathematics.  While my current interests lean towards more esoteric fields like quantum field theory, I am really most interested in finding a research area with interesting problems and interesting techniques.  In the little bit of self study in physics and math that I’ve done of the last few years, I’ve learned that seemingly different branches of physics (and math) are much more interconnected than I would have ever thought as an undergraduate.  The details of the problems may be different, but the tools and techniques developed in one field often seem to find good use in a seemingly unrelated field at a later time.  So in addition to exploring opportunities in particle theory, I would also spend my first year as a graduate student learning more about research in other theoretical fields being researched by faculty at Iowa.

In closing, I would like to point out that I have already begun the process of taking physics classes to warm up in preparation for the possibility of joining the physics department here at Iowa.  I sat in on Professor Rodgers graduate level General Relativity course last semester.  Though I was auditing the class (no grade or credit), I made sure to do most of the homework exercises because I knew that it would be difficult to truly grasp the material otherwise.  I also completed the midterm exam and did very well on the exam.  In addition, I am taking (for credit) the mathematical methods course taught by Professor Scudder this spring.

When I was growing up, my father would try to explain some aspect of the universe to me by drawing little diagrams on a napkin.  He wasn’t a scientist by trade, so what he knew of the universe came from reading popular science books.  I couldn’t tell you what was written on those napkins, I never really understood what he was saying.  It is likely that he wasn’t always sure himself.  The effect wasn’t so much to create a child who lived and breathed all things science and mathematics, but rather to instill in me a genuine fascination with everything about the world around me.  It was a gift that would take some time for me to fully appreciate.

By the time I arrived at the University of Maryland, just ten minutes down the road from where I grew up, I had little idea of what I wanted to study.  At the end of high school I had picked up some popular science books myself, several on quantum physics, but physics seemed to be a subject that really really smart people studied.  Maybe I was smart, but really really smart?  After having crossed off almost every single major from the university’s catalog, I found myself with two potential options:  English, a subject that I had thrived at in high school and physics, a subject that intrigued me, but which intimidated me thoroughly. Then, at an evening colloquium for freshman, I went to hear Dr. James Gates talk about his work in String Theory.  I was fascinated by the talk and at the end I approached him:  ”How do you know if you are smart enough to do physics?”  His response set me off on the path I’m on today:  ”You don’t, until you try.”

For four years after he spoke those words, I didn’t look back.  I dove into physics in a way I had never really done with anything else before.  I found another gear in my work ethic, in my dedication and seemingly in my ability to think.  I also found a community of people who loved this subject as much as I did, who were willing to work late into the night to get that last problem right.  Those were some of the best four years of my life.  And all the while I began preparing for the next stage, studying physics at MIT or Cornell or Columbia, schools that as a senior in high school had held no allure for me whatsoever (nor, as a solid A-B highschool student with average SAT scores would I have held any particular allure for them).

But during those four years I was also developing in other ways.  I was starting to feel a pull to do something for the world around me that wasn’t quite so esoteric as studying particle physics.  While I loved physics, I felt the need to expand my horizons in other ways as well.  So I made the decision during the fall of my senior year to forgo graduate school for a year or two and apply to Teach-For-America, where I would (in my own mind) help lift young school children out of poverty by my sheer enthusiasm for physics and math.  Unfortunately for me, and perhaps fortunately for those children, Teach-For-America did not believe I would be able to accomplish such an ideological feat.  And so it was that I found myself at the end of my senior year without a plan for the coming year.

What began as a year or two hiatus from physics turned into three, then four, five…now almost ten years later I have moved around the country several times, studied tropospheric ozone, spent a month hiking in Alaska, gotten married, bought a house and written over a hundred patents.  And all the while I have felt this nagging feeling that I’m not done with physics and it isn’t done with me.  I have spent some of my free time teaching myself quantum mechanics and field theory.  But the results have been mixed.  I still have that same strong work ethic I had when I was an undergraduate (see the > 100 patents written in the last few years), but I’m missing so many other crucial ingredients:  a strong community of other people who love and are devoted to physics, the structured progression of courses and the direction of professors and advisors.

I want to pursue a PhD in physics here at the University of Iowa because I realize that there are questions about the physical world around me that I am fascinated by, and they are questions that I could be a part of answering.  I’m not content to read popular science books.  I want to be on the front line.  Whether it be in plasma physics or quantum field theory.  As an undergraduate I was overwhelmed by the thought of doing research myself, I was just happy to learn anything I could about thermodynamics or electromagnetism.  I felt lucky just for the chance to observe.  But now I feel more confident than ever that I bring something to the table as well – that I have something to contribute to physics.

It is true that my exposure to research topics to this point has been extremely limited – mostly I know what I’ve read in popular accounts of cosmology and particle physics and the little bits I’ve picked up in my own self study of quantum mechanics, field theory and mathematics.  But I’ve also come to realize that physics is much more interconnected than I’d ever realized as an undergraduate – that many of the elegant themes I’ve encountered in esoteric topics like quantum field theory have applications in condensed matter physics and vice versa.  It could be that the techniques needed to solve a problem like quantum gravity will come from some seemingly unrelated field like nonlinear dynamics or quantum electronics.  So while my inclination is towards particle theory, I don’t feel the need to pick a particular research area at this early stage.  Instead, should I be accepted to the program at Iowa, I would spend time during my first year seriously looking for a research area with interesting problems and interesting techniques.

While my primary interest in pursing a PhD in physics is to become a researcher, I am also very interested in teaching physics.  As a patent writer for the last several years, my primary responsibility has been to explain (using words and figures) complex technical inventions in a way that can be comprehended by someone without expertise in the technological field of invention.  Furthermore, I have a responsibility to boil down the invention to its essence – some generalized features that can be modified in a myriad of ways without changing the original intentions of the inventor.  I believe that these are two qualities that a good teacher should possess: to break down a complicated subject into more easily understandable pieces and to highlight those essential features, the motifs, that provide a backbone for more sophisticated topics down the road.  I am excited about the prospect of teaching other students not just the formulas and laws, but the way of thinking about problems that is common to much of physics.  I believe that one of the greatest things that I learned from my undergraduate studies at Maryland was that basic physics trick of taking limits – considering what the solutions would be at some extremes, and then deducing what the solutions might look like somewhere in the middle.  As someone who routinely struggles to do basic arithmetic in his head, I have used this limits trick countless times over the last ten years of my life.  It is a certain way of looking at the world that becomes part of you once you have seen in work successfully in solving problems in mechanics, E&M and quantum mechanics.

Let a and b be numbers such that a < b.  Then every sequence in the interval [a,b] has a subsequence that converges to a point in [a,b].

Before stating the proof, we consider a key lemma:  Every sequence has a monotone subsequence.

Proof of lemma:  Consider a sequence .  We can a natural number m a peak index for the sequence provided that for all integers .  Now, we can have two possibilities:  (1) there are finitely (including 0) many peak indices or (2) there are an infinite number of peak indices.  Assume (1), then we can choose an N such that for all n greater than N there are no peak indices.  But that means, by definition of a peak index, that choosing some  for n greater than N, we can always find another element that is larger.  And we can do this indefinitely, so we can build a string of numbers, each greater than the last.  This gives us the desired subsequence (increasing monotonically).  If, instead, we assume (2), then we have an infinite number of peak indices, each of which (again, by definition) is smaller than the previous index.  So we have a monotonically decreasing subsequence.  

Another lemma:  Every bounded sequence has a convergent subsequence.  Well, we know by the preceding lemma that every sequence has a monotone subsequence, and we know from the monotone convergence theorem that every bounded monotone (sub)sequence converges, so the lemma is proved.

Ok, now to the proof of the main theorem.

Proof of BW Theorem:  Let  be a sequence in [a,b].  Then  is bounded.  By the preceding lemma, there is a convergent subsequence of .  Now, since is a sequence in [a,b], the limit is also in [a,b] (by another result in Fitzpatrick which I won’t state or prove here).

That is it.  Pretty simple, really.  I’ll admit the concept of a peak index wasn’t completely obvious to me at first. I mean, it is straightforward the way it is defined, but I wanted to come up with an intuitive picture (a graph of a function, for example) of why it would have to be true.  I don’t really have that picture.  But I’ll keep working on it and see if I can’t come up with something later on.

Am I totally crazy to think I can teach myself advanced mathematics without a program, fellow students and a graduate advisor?  In some sense, yes, I am totally crazy.  But on the other hand, I think there are some pros to doing things this way.  I haven’t gotten them all figured out yet, but for starters, I’m not sure I am in a position to get into a graduate program in math at this point, at least not given my current geographic location and lack of formal training.  So, if math is what I want to study, I don’t have a lot of options.  Besides, I revel in the thought of doing things my own way, off the beaten path, even if that path is so far away from the beaten path that odds are no one will ever know where I went or know to come looking for me should I get lost, or should I discover some brilliant theorem.  But I feel that mathematics is more of an art form than anything else, and I will count myself among those starving artists out there who spend their days toiling away on works of art that may never see the light of day.  Also, I do believe that the internet gives me an advantage over someone in my situation (who would put themselves in this precarious situation, I’m not sure) from years earlier.  I will have the option to correspond with people on online forums.  Not that it is a substitute for an advisor and actual colleagues.  But it is better than nothing.

Anyways, I call it my five year plan.  I figure it will take a year to review basic undergraduate material, then two years to learn graduate level mathematics and another year to learn any more specialized material.  The last year would be my first chance to take a crack at actually solving some problems, a la a thesis of sorts.

Is this going to happen?  If my track record is any indication, probably not.  But I’ll be damned if I let that keep me from trying.

The monotone convergence theorem says that a monotone sequence converges if and only if it is bounded.  I mentioned in an earlier post that a convergent sequence must be bounded.  If a sequence is unbounded, it will not be possible to establish the convergence criteria, since some numbers in the sequence would slip outside of any arbitrary epsilon ball (if this only happened for a finite number of elements, a bound could be established).  Therefore, we need only prove that if a sequence is bounded and monotone, it necessarily converges.

Proof of the monotone convergence theorem:  Suppose is monotonically increasing.  Let .  Then S is bounded by assumption.  By the completeness axiom, S has a least upper bound.  Let a=l.u.b. S.  We want to show that the sequence converges to a.  Let .  We need to find N such that for all or for all .  Since a is an upper bound for S, we have for all n.  Also, since a is the least upper bound for S, is not an upper bound for S, so there is an N s.t. .  But since the sequence is monotonically increasing, for all .  Thus we have that for all , as desired.  So the sequence converges to a.   A similar argument can be made for a monotonically decreasing sequence.  .

Intuitively, the idea is that we have an increasing sequence that is bounded above (for example), thus the sequence must be getting closer and closer to the least upper bound.  Again, it is that idea that because the sequence is infinite, it has to keep increasing, and we can always get it a little closer to the intended target, which is the least upper bound.

Now, for understanding the Bolzano-Weierstrass Theorem, it will be useful to consider a few other theorems.  The first theorem states that every sequence has a monotone subsequence.

We recall that a sequence of real numbers is a real-valued function whose domain is the set of natural numbers.  I.e., a sequence is a function , typically denoted as .  Sequences can be defined explicitly, recursively, or another way.  We can also talk about the convergence of a sequence.  A sequence converges to a number a (called the limit of the sequence) if for every there exists an N such that for all .

Convergence has to be one of the most important concepts in modern mathematics.  My naive sense is that most of analysis is really built upon the notion of convergence (if, we grant that concepts like continuity, etc.  are themselves built on top of convergence).  Convergence allows us to talk about the behavior of an infinite number of numbers in a meaningful way.  Intuitively, convergence of a sequence means that we can get arbitrarily close to a number.  But moreso, that at some N, every element of the sequence is arbitrarily close to the number.  When I took my first course in analysis (almost 10 years ago now) I remember thinking of a convergent series as having a “bob” and a “tail”.  The “bob” is the cluster of numbers beyond some number N, where N is arbitrary.  Every number in the cluster is within some arbitrary epsilon of the limit.  The tail is made of a finite number of numbers, at least some of which are further than epsilon away from the limit.  Of course, the sequence may not actually look like a bob and tail when plotted, but I think it is a useful picture.  Give me some epsilon, and I will find a large enough N so that I can produce that bob and tail.  Then I can just lob off the tail and talk about properties of the sequence since I know that every element is in some epsilon ball of the limit.

Now not all sequences must converge.  Take the sequence of natural numbers.  These numbers do not converge for sure.  What sorts of properties might we want to have in a sequence to at least give us some hope that it will converge?  Well, for starters, we would like the sequence to be bounded in some way.  The problem with the natural numbers, for example, is that they run off towards infinity.  If we imagine drawing an epsilon ball around a sequence, if the sequence is not bounded above and below, it will eventually make its way out of that epsilon ball and ruin our notion of convergence.  So we consider the following definition, which will prove to be useful.

A sequence is bounded provided that there is a number M such that the absolute value of each element in the sequence is less than or equal to M.

At this point, one can go on to prove various properties of convergence, such that every convergent sequence is bounded.  Again, this must be true, otherwise elements of the sequence would leak out of our imposed epsilon balls.  Now the converse need not be true, since the sequence {0,1,0,1,…} is bounded but does not converge to a number.  You can also prove things like that if a sequence converges to d and every element of the sequence is greater than or equal to 0, then d must be greater than or equal to 0.

At some point, I would like to post a discussion of the Monotone Convergence Theorem, the Bolzano-Weirstrass Theorem and the Nested Interval Theorem.  Not only because these are important theorems, but because they highlight the sorts of reasoning used in analysis in general (even if they are, by todays’ standards, rather elementary).

Take, as a simple example, Newton’s force law F=ma.  How does it work?  Well, you plug in the acceleration of an object and its mass, and the equation tells you the amount of force acting on it.  Or, you could plug in the force and the mass and it spits out the acceleration.  Seems simple enough.  But how does it work?  We could give a further break down of acceleration, as the change in velocity with time, of mass, as a localization of energy, of forces, as the effects of various types of “fields” interacting with one another.  But at the end of the day, we are left with some other variant of this equation which takes some inputs and spits out an output.  And we may still ask, how does it work?

In some sense, the laws of physics, of chemistry and of biology, work because, well, they work.  Take, for example, F=ma.  Now, mathematically, there is no problem in me writing down F=ma+m.  That is also a valid equation.  But it doesn’t describe the physical world around us, so far as we know.  So in the physical sciences, there are a huge number of different mathematical equations we could write down, but only some will have some application in our world.  It is true that sometimes one equation can be explained by reference to some more “fundamental equations”.  For example, it is believed that the equations of quantum mechanics would, with considerable effort, ultimately yield the familiar force laws for classical mechanics.  In some cases, we can argue why law of physics should be the way it is.  Take for example, Einstein’s theory of gravitation, which shows us that acceleration and gravity are in some sense the same, and that gravity is just a manifestation of the bending of spacetime.  Since Newton’s theory can be derived as an approximation of Einstein’s equations, one may argue that we have given sufficient evidence why gravity is the way it is.  Except that there is a bit of circular reasoning going on here.  Einstein’s original idea for relativity came from the fact that all objects experience the same acceleration, which is a feature of Newton’s law of gravity.  Also, to “tune” his original equations, Einstein compared the results of his theory with the results given by Newton’s theory in certain circumstances.  So Einstein borrowed from Newton in building his own theory. In some sense then, Einstein’s theory doesn’t really explain WHY gravity is the way it is, it only gives a much more accurate theory.

And similar problems occur again and again in science.  New theories are usually built out of pieces of older theories.  So the more “fundamental theories” already have encoded in them some general principles that were assumed in the original theories.  It leads to a long chains of dependencies from which we can never really answer the question of “how does it work” in the idealistic way we’d like.

But that is not to say we are merely speculating and not making progress.  We can make predictions about the motions of planets using relativity that are much more accurate than those using Newton’s equations (at least, in many situations this is true).  There are phenomena that don’t appear in one theory but do appear in another.  There are certainly, objectively identifiable, “better theories”.  But why and how a theory works is usually a very complicated thing that ultimately must be answered with, “because it works.”  And that isn’t a bad thing. Logically, it is hard to fathom a theory that could be explained using some basic set of first principles.  Logically, these principles could conflict with one another, or they may be unable to explain all phenomena.

We shouldn’t forget that our brains operate not in the strict logical sense that we use to write computer code, but in a vastly more complicated processes that take inputs and give us outputs.  Our brain is the ultimate black box.  Not in the sense that we can’t explain how it operates, but in the sense that to give an explanation of how it operates (like giving an explanation of how F=ma works) is not necessarily going to clarify most problems.  It will clarify problems dealing with how our mind works.  But other problems would only become more complicated.

Sometimes, then, I think that the advice of the pragmatic scientist is useful.  Build a theory.  Test it.  If it works, try to make it better.  If it doesn’t, build another one.  But don’t get too bogged down in understanding “how it works”, unless the answer to that question will lead you closer to understanding the real world you are trying to model.  Because we can’t forget that on one hand we hand our models and on the other hand there are things “out there”.  And we just can’t assume that they are in one to one correspondence with one another.  So when we have a model that predicts how some system behaves, we can’t assume that the system actually works the same way as our model.  I think we spend, at times, too much effort trying to understand our models rather than trying to understand the real world.  Many times understanding our models will lead us to new insights, because in fact we have found that mathematics is a great approximation to the processes of nature.  But this only goes so far.  Mother nature always has new surprises up her sleeve and sometimes those surprises are impossible to find by taking apart our models and trying to understand all the pieces.  Usually, what is needed is more observation to point us in the right direction.

We can’t be fooled by the familiarity of forces, masses, pushes, pulls, etc., into thinking that Newton’s laws are somehow more “understandable” than the laws of quantum mechanics, for instance.  It is our complacency with Newton that leads us to this conclusion.  New scientific laws are like changes in a culture.  It takes times to adopt new behaviors and norms.  It may not be long before we are all claiming to have a deep understanding of the wave/particle duality, and questioning the philosophical grounds of some new and more bizarre theory.

In a similar manner, many of the problems in quantum mechanics that deal with measurement would seem to be solved by considering some larger (or at least different) system.  I’m not really sure that this actually works, because I don’t pretend to understand the measurement problem all that well, and because there are people arguing on both sides about whether or not the “measurement problem” is a real or perceived phenomenon.  My guess is that rearranging the systems under consideration solves some problems, but does not solve or even introduces others.

I believe that the problem of measurement could be served by viewing it in a logical manner, stripped of the particulars of the Schrodinger equation.  In addition, I think it may be useful to consider the sorts of measurement issues that may or may not arise in classical mechanics.

My intended purpose is not to invent some new “interpretation”.  I think interpretation is well and fine, but it is not physics unless there is prediction.  And it isn’t new physics unless there are new predictions made (or at least old prediction made with less obfuscation).  No, instead my intended purpose is to scrape away at the surface of the measurement problem and see if there are any real physical phenomenon that may give insight to the problem.  When Einstein started to consider moving trains and lightning strikes it could have been argued that he was simply worried about an interpretation of real phenomenon.  But what he was really after was any new phenomenon that had been obscured by the old ways of thinking about things.  The old rules, such as the Galilean additions of velocity law that proved to be wrong in the case where the velocities are approaching the speed of light.  Likewise, in considering relativity Einstein did away with the need (at least for some people) to speculate about the aether.  Perhaps there is some deeper physics lying in the measurement problem that would help to resolve it, or at least to elucidate some other perhaps related quantum phenomenon.

The first issue is to create a logical model that has similar characteristics to quantum mechanics.  Namely, equations that describe probabilities and objects whose behavior is described by these equations.  We aren’t concerned about the type of equation just yet.  We instead want to see how the most general systems behave.  Can we get a “collapse”?  Can we get something akin to decoherence?