Yesterday I caught the tube with a number of the post-docs and members of the faculty and we travelled to Imperial to hear Jose Barbon (Madrid) and Katrin Wendland talk as part of the London Triangle seminars which happen occasionally, and are shared between Imperial, KCL and QMW. These are usually the most technical of the different seminars I have attended, and the topics have a tendency to be quite specialist. Consequently I'm not going to feel too disappointed because I didn't understand much, and instead of describing the talks in any detail I'm going to give the titles and where the important papers can be found, and mention any vocab that caused me to lose my way.

Jose Barbon talked first on "QCD Chiral Dynamics and AdS/CFT Models", and it seemed he gave a very good talk and was very familiar with the various techniques of doing calculations using the AdS/CFT correspondence. Unfortunately I'm not so familiar with it and was lost quite quickly. The talk was based upon his paper with Hoyos, Mateos and Myers entitled "The Holographic Life of the \eta'" and new vocabulary was light on the ground, although I did hear the word "bolt" used for the first time, in this case it was described as being equivalent to a "wall repelling Wilson loops" (which was equally alien terminology for me), but the word itself does not turn up in the relevant paper.

After no questions, which I feel is often a sign that the speaker has bamboozled the audience, we enjoyed some very welcome coffee and sticky cakes.

The second talk was given by Katrin Wendland, who is based at Warwick and Chapel Hill, and was a very decent algebraic geometry lesson. Her talk was entitled "How to construct SCFTs associated to a family of smooth K3 surfaces" - first piece of vocabulary thanks to wikipedia:

KW's talk was based around her paper "On Superconformal Field Theories Associated to Very Attractive Quartics" and was concerned with expressing SCFTs in a neat form classified by a sum of quartic polynomials on CP^3. Incidently, if you look at the wikipedia entry on complex projective spaces you find it carries the health warning:

This article may be too technical for most readers to understand. Please expand it to make it more accessible to non-experts — without removing the technical details — and remove this notice once so done.I think whoever has the job of putting this quote in place should start doing the same job on the arxiv :) Anyway KW gave us a very specific recipe for characterising SCFTs as a Z_4 orbifold, from the tensor product of two Z_2 orbifolds. The example of the tensor product of two Gepner models was used and her paper with Nahm cited. The talk was very lively, and as with the Barbon, it was disappointing that I didn't know enough to benefit from it :(

I went back to Drury Lane with my tail between my legs and felt discouraged.

## 22 comments:

Hi Paul!

Looong time no see.

I take the liberty to postulate that comments are welcome at your Tangent Space; if not, I'll soon overcome the curved effects around it and once again be on my way. :)

I can't remember how I came across your blog a few days ago. A deep, beautiful, cosmic mystery indeed..

Reading, upon my return, your honest, literary input of yesterday, reminded me of the brilliant Woody Allen:

"Today I saw a red-and-yellow sunset, and thought, How insignificant I am! Of course I thought that yesterday, too, and it rained..."

but of the great Alexandrian Poet too:

"As you set out for Ithaka

hope the voyage is a long one,

full of adventure, full of discovery.

Laistrygonians and Cyclops,

angry Poseidon—don’t be afraid of them:

you’ll never find things like that on your way

as long as you keep your thoughts raised high,

as long as a rare excitement

stirs your spirit and your body.

Laistrygonians and Cyclops,

wild Poseidon—you won’t encounter them

unless you bring them along inside your soul,

unless your soul sets them up in front of you.

Hope the voyage is a long one.

May there be many a summer morning when,

with what pleasure, what joy,

you come into harbors seen for the first time;

may you stop at Phoenician trading stations

to buy fine things,

mother of pearl and coral, amber and ebony,

sensual perfume of every kind—

as many sensual perfumes as you can;

and may you visit many Egyptian cities

to gather stores of knowledge from their scholars.

Keep Ithaka always in your mind.

Arriving there is what you are destined for.

But do not hurry the journey at all.

Better if it lasts for years,

so you are old by the time you reach the island,

wealthy with all you have gained on the way,

not expecting Ithaka to make you rich.

Ithaka gave you the marvelous journey.

Without her you would not have set out.

She has nothing left to give you now.

And if you find her poor, Ithaka won’t have fooled you.

Wise as you will have become, so full of experience,

you will have understood by then what these Ithakas mean."

Best,

Dimitris

Yikes, you're supposed to learn string theory by your second year? At my university, which is admitedly one with a weak physics department, that's when students learn quantum field theory!

Hi Anonymous,

To be honest we're expected to have completed a string course before starting the PhD. But only the very basic stuff, i.e. part one of Zweibach, plus some SUGRA. I think that's fair.

I don't think we're expected to learn all of string theory until we're in our fifties :) and that's if the literature grinds to a halt today.

Best wishes,

I don't know how it's like in England, but are undergraduates supposed to know general relativity and supersymmetry (never mind SUGRA) and enough QFT to learn some basic string theory???

I'm been in an American physics department for many years now but my knowledge of QFT is still very shaky and apparently, that is true for the other graduate students who have been here for years as well.

I did a three-year physics undergraduate degree, which consisted of all the usual courses: E.M., thermodynamics, Q.M., special relativity, statistical mechanics etc. but no G.R. Then I took a one-year masters at Kings in mathematical physics in the maths dept. This is where the discontinuity occurred. In two semesters we were taught the basics of QFT, GR, SUSY and String theory, and some other topics. Don't get the wrong idea though, I don't think this course structure is average in the UK. Also don't forget that PhD's are funded for three years here, so some speed is needed. Consequently I'm terrified of renormalization theories, and other topics that I haven't practised enough, or at all.

That said, you don't really need too much background to study string theory in the light-cone gauge. Just look at Zweibach (I should be getting commission).

Hi Dimitris,

Indeed long time no see. How are you? & where are you? I'm guessing you didn't take the MSc at King's, so what are you up to now, besides raising the quality of writing in my blog with your well-chosen quotations. Thanks for those, I felt very proud that I have such an educated friend. I especially enjoyed "Ithaka", and will take it to heart, but as you know Poseidon is everywhere in physics, and it is enough to stay afloat by ignoring those things we didn't learn completely or forgot or never-saw-before, but: I don't like it! :)

Best wishes,

Thank you for your kind words Paul.

I'm currently spending my time at the uneventful, one-dimensional world of examinations! (i.e. eyes-->lecture notes!)

I'm at Royal Holloway completing my degree - which seems to be taking for ever (I am repeating due to "externuating circumstances" of last year - had to go back home).

I still hope to be able to go to King's this September, for the MSc (or rather, for the "Master of the Universe", as people used to call it), and do some postgraduate study there.

Well, Poseidon may be all around us, yet all we need to be, is the Odysseus that we are!

Best wishes!

:)

(dimitriospappas@gmail.com)

When I was in grad school, I ended up mostly reading a lot of review papers and lecture notes from various summer schools (ie. Trieste, etc ...) on various aspects of string theory. Occasionally I read the original research papers if there was either no review articles about a particular topic and/or the review articles were not very clear. These were usually the highly cited string papers like the ones by Gross, Witten, Schwarz, etc ... from the mid 1980's.

After awhile you'll have an idea which papers are important and which ones are marginal. The majority of research papers in any field, are mostly marginal ones. The important papers are usually the ones which have many citations (ie. in the hundreds, if not thousands). There's no point in reading most string papers for the most part, other than the ones directly related to the research problem you're working on. I came to the realizaton that most marginal papers were a waste of time to dwell on, for the most part.

Just wondering. What is the typical age when somebody gets their physics PhD in the UK? I did my physics PhD in America and finished when I was 27. Though in my case I didn't take any time off from high school to the end of grad school.

For a dedicated UK student, 4 years undergraduate (including a masters) and then 3 years earning their PhD would mean they would be about 25 years old. However many students here are from overseas, and they tend to have spent longer in their undergraduate degrees and consequently have a much more thorough grounding and general mathematical knowledge. I'm really jealous of the US system: I would like to have had a chance to go over my undergraduate work in more detail. However, I do like the speed of getting to the really interesting material that the short PhD course allows.

In my case, I took three years out, so I hope to be 28 when I finish.

Best wishes,

Paul,

If you don't mind me asking, what did you do in those 3 years you took off from doing academic physics?

What exactly makes you "jealous" of the American system? I think you may have an overly "romantic" view of it.

In the American system, an average student can finish undergrad in 4 years. Many take longer from either flunking one too many courses without being thrown out, and/or they do too much partying and drinking booze. (ie. President George W. Bush allegedly spent 6 or 7 years to finish a history degree at Yale). Occasional a really hardcore dedicated student can finish undergrad in 3 years.

Physics grad school resembles a combined masters and PhD at many major American universities. Many people take around 5 years or so, to finish a physics PhD in a theoretical area. (The masters degree is usually just a "token" awarded along the way). The courses I took in my first year of grad school were largely a review of similar courses I took in undergrad, except maybe grad level electromagnetism which used that dreaded Jackson textbook. Then we had to write the comprehensive/preliminary exam, which was largely used to "weedout" the slackers and weaker students.

For an American undergrad degree in physics, many places don't really offer advanced courses like general relativity, relativistic quantum mechanics, quantum field theory, string theory, modern condensed matter theory, group theory, etc ... at the undergrad level. (These are usually graduate level courses at many places). Though a hardcore physics undergrad who is able to finish off early courses like undergrad level classical mechanics, electromagnetic theory, nonrelativistic quantum mechanics, etc ... before their senior year, would be able to take the masters level graduate courses or even quantum field theory in their senior year. The people who decide to major in both math and physics, may not have the time and space in their schedules to be able to squeeze in many graduate level courses. (Something like quantum field theory alone is already a heavy course by itself).

In my case I didn't take any graduate level courses when I was an undergrad. The only "advanced" course I took in my senior year was a particle physics course, beyond undergrad level classical mechanics, electromagnetic theory, quantum mechanics, and statistical mechanics. (Though this particle physics course didn't do anything like calculating Feynman diagrams. It was taught by a particle experimentalist who mainly covered it from an experimental point of view, where we used that awful Perkins particle physics textbook.)

When I was an undergrad, I think I spent too much time goofing around and doing things like playing too many video games, going to rock concerts every week, playing too much pool, etc ... So I'm not a really good example of a "dedicated" physics student. The most I did resembling "dedication", perhaps was that I had several books on quantum field theory, particle physics, and string theory which I slowly worked through over those years when I had some free time. At the time I always wanted to know how to calculate Feynman diagrams, despite my low level understanding of the mathematics involved.

In hindsight I think I could have spent more time trying to REALLY understand the stuff covered in the undergraduate physics courses I took. All I did in undergrad was mostly just crank out a bunch of extra problems out of the textbooks, and attempting to second guess what the professor could put on the exams. It was years later in grad school after passing the physics comprehensive/preliminary exams, that I came to the realization that course grades mattered very little anymore. Even today I still don't think I really understand what quantum mechanics and quantum field theory are really all about, other than working out a lot of textbook exact solutions and perturbation theory problems.

Years later after I finished my PhD, I went back through some of my old undergrad textbooks and books like the Feynman lectures, Landau/Lifshitz, etc ... largely out of boredom. When I look back at what was covered in undergraduate physics, I came to the realization that I largely "glossed over" a lot of the intricate details and subtleties involved. It took me a long time to appreciate a lot of these deeper details and subtleties. Since today I'm not academic anymore, I end up spending a lot of my free time trying to understand these things. I got the impression a lot of my classmates from undergrad and grad school, at the time also didn't really appreciate these intricate details and subtleties in physics.

I think what you may be "jealous" about concerning the American system, may very well be something that most American physics undergrads don't even have a clue about. I suspect most physics undergrads in America, UK, europe, Japan, China, India, etc ... may very well all be clueless about the deeper intricate aspects of physics. Perhaps this is more of age and/or maturity thing, than the actual educational system itself?

On the other hand, British colleges don't have a liberal arts policy for undergraduates, so I guess they can cram in a lot more physics in 4 years than Americans.

Dear anonymous,

Thanks for your detailed post about the US system. Perhaps I should have been a bit more specific. I meant to imply that I was jealous of the PhD system in the US, which, as was pointed out, commences with a 2-year review of "foundation" physics.

At the start of my PhD, last year, I was very aware that my taught education, in the main, had come to an end but, I hadn't really fleshed out some very important ideas. For example renormalisation in QFT. Nor had I seen a reasonable justification of some of the more esoteric aspects of my masters courses, such as the "sum_{p>0} p=-1/12" in the reordering of the virasoro operators of string theory :) So when I said I was "jealous" it was because I was imagining the US PhD system would have given me plenty of opportunity to chase up any questions I had from earlier courses. Perhaps the grass just seems greener, but there is a lot to learn, and 3 years of PhD study to catch up on 25 years of research (for supergravity) leaves little room to recapitulate one's undergraduate study.

For my 3 years out, I tried my hand at a variety of jobs, none of which involved any maths, nor were quite satisfying enough. Specifically, I worked as trainee "pay consultant" (assessing the roles of company directors and benchmarking and reporting on how much the median of a group of their competitors would be paying for the equivalent services, so they pay themselves a competitive amount), as an office temp for the Learning Disabilities' team of the Social Services in Tower Hamlets, and finally as a junior software programmer for a radar company on the Isle of Wight. Because I wasn't mentally stimulated, I found myself reading Landau and Lifshitz vol. 1, in the evening, and Relativity by Rindler. And subsequently I took the masters course at KCL, and haven't looked back, well, until now :)

I should say that my mathematical thinking after not doing any maths for three years, had surprisingly improved. I think this was because I was taking study much more seriously (although I can't quite kick the idea that many ideas feel into place when I wasn't concentrating on them), and also feel that a "mature" approach to study is very important, so long as the course has a reasonable structure. But there is a difference between a 3-year and a 5-year PhD, and ideally I would prefer an extra year on the UK PhD. I'm sure I will believe this passionately next year when I have to write up :)

Best wishes,

- I agree with the last anonymous (at 5:10 pm) poster that many American universities require a number of liberal arts and/or general ed courses as a part of the undergraduate degree requirements. At some places, these requirements can literally stretch into almost the equivalent of a whole entire school year. Perhaps this is why American undergrad degrees usually take up 4 years, compared to the British 3 year undergrad degrees?

It seems like in America, the most popular courses for satisfying the liberals arts and/or general ed requirements, are freshman and sophomore psychology courses which seem to frequently have a lot of "multiple-guess" type of exams with no essays required. I remember in some of these psychology multiple-guess exams I found kicking around the dorms, the last question on some of these exams was something amusing like "make up five new multiple-choice questions which are not on this exam". In some cases the professor would actually use some of these student "made up" multiple-choice questions on future exams!

- In reply to Paul's previous post, I'm not sure if a 5 year American PhD program is that drastically different from a British masters + 3 year PhD program.

The first year of an American physics PhD program is mainly taking courses and preparing to write the comprehensive/preliminary exams. The first year graduate courses we had to take were: classical mechanics (using Goldstein's book), electromagnetic theory (using Jackson's book), quantum mechanics (using Sakurai's book), statistical mechanics (using Reif's book and Plischeke & Bergersen's book), and mathematical physics (mainly a review of complex analysis and partial differential equations). The comprehensive/preliminary exams were usually 7 hour exams which covered stuff at the level of these first year graduate courses. For the most part these courses were largely a "review" of what I saw in equivalent senior level undergrad courses. (There's a very large significant overlap between these first year graduate courses and equivalent senior year undergradate courses). The only "new" things which I didn't really see previously were perhaps things like the Dirac equation and the corresponding solution to the hydrogen atom, the exact solution to the Ising model, and some of the solution methods in Jackson's electromagnetic theory book which were glossed over in equivalent undergrad E&M courses (usually using David Griffith's undergrad E&M textbook). In principle these "new" topics could be incorporated into equivalent senior undergrad courses, or one could just learn them by self study. Over the years I noticed a lot of folks who did their undergrad over in Europe and/or in some places in Asia, actually used these same textbooks (ie. Goldstein, Jackson, Sakurai, etc ...) in their equivalent senior year undergrad courses. What textbooks were used in your senior year undergraduate courses?

In principle the American first year physics graduate courses can all be done by a really hardcore dedicated student in their senior undergrad year, if they're able to finish off all their undergrad physics courses (ie. classical mechanics, electromagnetic theory, quantum mechanics, statistical mechanics, mathematical physics, etc ...) in their sophomore and junior years (2nd and 3rd year repsectively). If one is able to pull this off, then in principle one can finish an American physics PhD in four years instead of five.

The second year of an American physics PhD program will usually involve taking more advanced courses like quantum field theory, general relativity, particle physics, string theory, supersymmetry, etc ... for somebody wanting to do particle theory and/or string theory. Your description of the masters degree program in mathematical physics at Kings, sounds almost identical to what many 2nd year physics grad students go through at many American universities.

Many theory grad students will start their research around the end of their 2nd year or the beginning of their 3rd year, in many American PhD physics grad programs. After the first two years of taking courses and writing exams, most theory grad students only have 3 years left to finish their PhD research and write up their thesis. Frequently the grad student funding is only guaranteed by the department for 5 years maximum (usually in the form of teaching assistantships), unless one's thesis advisor is willing to fund their student for longer after the initial 5 years are up.

Paul,

I can relate to the feeling of being "clueless" about cutting edge research things as well as the deeper aspects of quantum field theory, when I was in grad school.

After I took courses in quantum field theory and particle physics, I still didn't feel I really understood what renormalization was all about. It took me subsequently several years of doing some deep thinking and extensive calculations before I had a better handle on renormalization. I ended up trying to do many different calculations like one-loop corrections to various propagators and vertices, as well as two-loop corrections to see how the overlapping divergences worked out.

When it came to supersymmetry and supergravity, I ended up reading and working out many of the calculations in the supersymmetry review paper by Sohnius. (I never took any courses on supersymmetry or string theory, since they were not offered). Later I looked at the supergravity review paper by Van Nieuwenhuizen, though I didn't attempt to work out every single calculation in it. (Some of them looked really nasty to do in component notation!) Years ago there were two huge books of reprints of major supergravity and supersymmetry papers: "supergravity in diverse dimensions" (ed. Salam & Sezgin) and "supersymmetry" (ed. Ferrara). Whenever I had to look up something about supergravity or supersymmetry, I usually went straight to these two sets of books. (Each section in these books had a 3 or 4 page summary of the major results of several papers). There's a more recent set of lecture notes on supergravity by de Wit at http://arxiv.org/abs/hep-th/0212245

Besides the 11 dimensional case and some of the 10 dimensional cases, most other supergravity cases aren't really all that necessary to know immediately unless you're working directly on a problem which requires specific knowledge of one of these cases. (ie. Doing standard model and/or grand unified theories with supergravity corrections usually requires knowledge of the 4 dimensional supergravity case). I always thought that supergravity in less than 11 dimensions looked like a huge mess.

For string theory, I ended up reading and working out many calculations from Frampton's "dual resonance models" book. (In the early 1970's, string theory was also known as dual resonance models). After reading and working through Frampton's book, a lot of the 1980's string theory papers and Green/Schwarz/Witten's books were a lot more clear to me.

I don't believe one can acquire an in depth knowledge of a particular subject, from just taking courses alone. Over the years I found that most of my in depth knowledge of various subjects, largely came from thinking extensively about a particular subject and working out some of the detailed calculations to convince myself of their validity. Otherwise things would have be taken on faith and/or "authority", which is not my preference for understanding. ("Proof by authority" doesn't really work too well after a certain point.)

"Proof by authority" is probably unavoidable. At some point, unfortunately :), one has to avoid the compulsion to check past calculations and work towards a result, trusting the experts along the way. Thanks very much for your references, I will be looking them up shortly and ticking off some of my to-be-understood list :)

Best wishes,

Thanks to all the anonymous readers relating their tales of coming to grips with studying theoretical physics. It is notable that all comments were positive and constructive, which is very pleasing.

My utopian idea of the US PhD has been redressed, and I now imagine it's not so different from the shorter PhD in the UK after all. I didn't realise that topics like SUSY, strings, QFT etc weren't studied until the second year.

My final year undergraduate texts (from a physics degree) included, Solid State Physics by Kittel, Statistical Physics by Guenault (I remember this being a good read), Nuclear and Particle Physics by Williams and Introduction to Mathematical Physics by Chun Wa Wong (I stuck to this over the Boas text because it was smaller to carry around).

My masters year texts, included The Large Scale Structure of Spacetime by Hawking and Ellis; D'Inverno's Relativity book; Weinberg's Quantum Theory of Fields (vols. 1 and 3); Green, Schwarz and Witten vol. 1; Polchinski vol. 1; Introduction to SUSY and SUGRA by Peter West; An Introduction to Hilbert Space by Young and Matrix Groups by Baker, amongst others. I still use these books now.

So there was zero overlap between my final year undergrad texts and the masters' books. Also I'm not familiar with any of the Goldstein, Jackson, Sakurai books that were mentioned.

Best wishes,

Paul,

I agree that some things may very well have to be taken on faith and/or "authority", in order to get things done in a reasonable amount of time. I never tried proving all the theorems about Banach and Hilbert spaces. Nor did I try to work out many of the nasty long supergravity calculations, nor any of the renormalization schemes for electroweak theory beyond 1-loop.

The most I ever tried to work out for 2-loop calculations, was for phi^3 theory in 6 dimensions. (This was the easiest case to see how the overlapping divergences worked out, with the least amount of complicated baggage). I tried working out some of the two-loop calculations in quantum electrodynamics, just to see how the overlapping divergences worked out, though I never bothered to go much further than that. I never tried to work out the 2-loop correction to the QCD beta function. It seemed kind of pointless, unless I was directly working on problems related to higher order loop corrections to QCD.

Paul,

Over the years I've noticed the folks who could do things like quantum field theory, string theory, etc ... in their sleep before they started graduate school, were typically folks who really studied advanced math and physics when they were a kid or a teenager. I knew one guy many years ago who could do multivariable calculus when he was in the 4th grade, and he knew how to do classical mechanics and electromagnetic theory problems by the time was in the 6th grade. Before he was even finished high school, he was already doing things like algebraic topology, quantum field theory, general relativity, etc ... By the time he was finished undergrad, he already had three particle theory papers published. I have no idea what happened to him afterwards, considering I haven't seen his name on any physics papers ever since. (I presume he's not in physics anymore. Maybe he got bored of particle theory?)

Most other folks I knew over the years, never heard of physics or calculus until their senior year of high school, when they first had to take these subjects. In my case, I found high school quite boring. I ended up spending more time skipping school and hanging out with friends who were into drinking booze all day. Only reason I majored in physics in college was that I read several articles about particle physics where I thought Feynman diagrams looked cool, and wanted to know how to calculate them. What was your main reason for majoring in physics in university?

It's more than a year. Most liberal arts colleges also require a minor and I think the other liberal arts requirements take up about a third of the required credits.

I like your reasons for studying physics :) My reason came when I started reading a lot of popular science books at about 14, things like A Brief History of Time, In Search of Schrodinger's Kittens and so on. Although I didn't really understand a lot of them I couldn't escape the fascination that there were people in the world who actually believed in tunneling, teleportation, black holes, parallel universes and so on. And I wanted to know why.

Paul,

I never really came across that many popular science books when I was a kid or teenager. The only ones which I read from cover to cover, were a series of books titled "understanding physics" by Issac Asimov. Most of the particle physics articles I came across were from Scientific American magazine.

Before I started my freshman year of university, I went several times to the local university library and tried to read several textbooks on particle physics. Besides reading the usual descriptive first few chapters, I quickly got lost once they started to do the Feynman diagram calculations. The most I was able to figure out at the time, was combining a bunch of physical constants and other relevant variables (ie. masses of particles, etc ...) and getting something which had the units of a cross section. At the time I was wondering why the Feynman diagram calculations were so tedious in obtaining some final numerical constant, when the final form of the cross sections could be otherwise guessed just by simple dimensional analysis. Despite being somewhat naive at the time, I was fascinated by how a lot of physics could be determined from just calculating a bunch of Feynman diagrams.

It was years later when I came to the realization that one had to have a lot of prior physics knowledge, before Feynman diagrams actually made any sense, and that the diagrams were a lot more than just a bunch of cool looking "pictures". During my freshman year of university, I tried working out some of the quantum mechanics textbooks problems and calculations on my spare time. Despite getting most of the correct answers, I didn't feel I really understood what the underlying physics was all about. Despite not understanding the underlying physics, I went ahead and tried to work out some basic quantum field theory calculations. (Most of them were problems on second quantization of free scalar and free Dirac fields). Despite getting most of the correct answers, everything looked like a huge haze of math calculations to me, without much physical understanding on my part. (It felt somewhat "robotic" in doing a lot of long calculations, and not really understanding why one is doing them). It took me another few years of physics courses, before I started to understand what exactly I was calculating.

By the time I was finished my freshman year of college, I felt I had a good enough grasp of how to do a lot of long calculations in various advanced areas like quantum mechanics, quantum field theory, general relativity, etc ... My main shortcoming was in understanding the underlying physics of what I was calculating. The rest of the undergrad physics courses I took were mostly helpful in understanding the underlying physics and conceptual ideas, which eluded me previously. At times I wonder whether I would have made a better mathematician than physicist.

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