How do you remember it all at once?

[Cruikshank]

Background: BS physics MIT, 4 attempts at graduate school, including Ph.D. math, Ph.D. physics, and MA Math (which I finally got, mostly to demonstrate that I can finish things.) 20 years as a professional tutor, full time, math and physics etc.

When I was 6 I knew I wanted to be a scientist. By age 10, it was physics. By age 12, it was fundamental theoretical physics. At age 28 I gave up. Apparently I'm not smart enough to be a physicist. Not being rich, being shy, and developing chronic disabling hand pain was just three strikes against me.

I review linear algebra, and find I've forgotten complex calculus. I relearn that, and differential equations are rusty. I practice that, and I forget differential geometry. I relearn that, and the linear algebra is rusty again.

Part of the problem is that I never get to USE any of this. It's as if I were studying a dozen foreign languages while never being allowed to speak any of them. My job allows me to stay sharp on all the basics: calculus, vector calculus, basic diff eqs and basic linear algebra and probability and statistics. No problem.

How do you actually get all the way to an interesting problem? I have lots of questions, curiosity about physics, but if I actually try to read something, I find that I must first study some subject as a prerequisite. Then that prerequisite has a prerequisite. Then that has two prerequisites. Eventually I find a place to make progress, but I'm so far away from anything interesting that I only get anywhere via tedious grind. I left graduate school because I felt that I was losing my MIT education faster than graduate school was adding anything, and decided that however badly I would do on my own, it couldn't be worse than grad "school," which felt like an active impediment to learning any physics.

I'll die of old age before I feel ready to learn QFT. I own hundreds of textbooks and have probably mastered chapter 1 of half of them, some farther. I feel as if I can ace anything facing freshmen, and have to struggle with anything more advanced.

I have worked nearly every problem in the first 6 chapters of Bransden and Joachain (graduate QM, though it felt about equal to my undergraduate QM at MIT.) I had to teach myself generalized distributions in order to make any sense out of the sloppy Dirac deltas used. I have worked through nearly every page of Shilov's Linear Algebra. I filled three binders with notes, expanding Spivak's Calculus on Manifolds into something comprehensible by mere mortals. I have put on my hip waders and slogged through philosophy of physics. I taught myself optics, electronics, the first four chapters of Group Theory by Hammermesh.

But I quit Hammermesh, for example, because it was some of the dullest and most tedious junk I've ever seen--despite the chapter titles sounding exciting! I keep seeing that--every physics and math subject seems to be a mix of the absolutely fascinating and the mind-numbingly boring. I drag myself through the boring parts for the sake of the good parts, but sometimes, the good part was just a mirage, or the carrot moved, putting another prerequisite in the way.

I teach for a living. I help other people learn, get them past roadblocks, show them what they need to do. But just as doctors make the worst patients, I find that watching a typical physics lecture, they might as well be writing the equations with their fingernails. Years ago I resolved NEVER to try to learn math from a physicist ever again, for sanity's sake.

I know that by actually talking to physicists and physics students (more than I ever managed while I was in college), I am probably endangering my qualifications as a crackpot. But I really do want to know...how do you do it?

How can you stand to wade through the boring bits?
How do you ever GET to the good bits?
How do you ever get to USE any of this material?
Am I just completely unsuited for physics? And if so, how was I supposed to figure that out?

I know my fate is most probably to be a dabbler for the rest of my life. I have hopes of writing textbooks to save others some of the grief I had to go through. I'm told I have a gift for teaching. But I really don't know what was so wrong with me, that I couldn't be a physicist.

 

[ahsanxr]

I understand your struggle, and though I'm less experienced than you, I can relate to the feeling of being stuck with learning prereqs. I think the important piece that you are missing, without which everything seems disconnected, boring and hard to remember is "unification". What I mean is that what makes things truly click for me is making connections between what I've already learned and internalized, and what I'm trying to learn. For example, you say you've learned Linear Algebra using a pure math book. It's still entirely possible that even after that, the first chapter of Shankar may seem foreign, since the notation and terminology used is completely different. What made things clear to me is making sort of a (mental) translation table which translates bras, kets, inner products and completeness relations and how physicists use them to the linear algebra I had already learned. If I didn't sit down and make this effort, it would be like "Oh f*ck I just waded through a semester of difficult proof based LA, but this physicsy version of LA still makes no sense. I guess I'll just have to relearn it the physics way. I'm stuck forever learning the prereqs, will I ever get to QM?"

The same process holds for differential geometry and general relativity, representation theory and quantum field theory, etc. An important thing not to miss is to treat everything as building upon each other. As another example, say you wanted to learn differential geometry properly. Among the first things you would learn would be things like differential forms and tensors etc. However a proper intro to these things heavily uses concepts from linear algebra. Similarly, theorems from multivariable calculus and analysis will show up frequently. So in your effort to learn diff geo, you've refreshed your memory of analysis and linear algebra. Now you think you have a good idea of diff geo. Move on to Wald or Carroll and learn about general relativity. Again, you make the effort to connect your understanding of diff geo to how Wald or Carroll present it and as a result your understanding of diff geo is also enhanced, and you'll get to use it in exciting ways when you do GR. This will also make GR click better, since you've now formulated it in a language you were already fairly familiar with, from your previous study of diff geo.

So the important thing is to always try and keep making connections. Do something like drawing big charts and try to see where everything fits in the bigger picture. This process is what keeps learning math and theoretical physics an enjoyable process for me.

And once you get to an advanced enough topic, since you'll be using a lot of these tools simultaneously, they'll be pretty much ingrained to your head:

Let's say you were doing a calculation in quantum field theory. You start with the lagrangian, and attempt to deduce the Feynman rules. Finding the propagator, which is the inverse of a differential operator acting on a space of functions (linear algebra), will have you solve a differential equation (by using Fourier transforms, greens functions). After you have those, you attempt to do a calculation to a certain order in perturbation theory. This will require doing a ton of integrals, some of which will certainly involve doing some contour integration (complex analysis). And of course, you have to remember that at the end of the day, you're working with states and operators which live on the Hilbert space of your QFT (again, linear algebra). And if you're dealing with a theory like QED or non-abelian gauge theory, or basically any relativisitically covariant theory, you'll also have to know your tensors well too.

 

[ModusPwnd]

Am I just completely unsuited for physics? And if so, how was I supposed to figure that out?

I figured it out when I wasn't able to get a PhD. Honestly, I think its silly to pursue something that you thought of when you were 12. You were just a kid, you didn't know crap about the world and the world of work. Most people go through many changes between 12 and being an adult. Now you are an adult and can choose a number of things. I work in a restaurant now, the pay is low but its something I can do. There are many choices out there, you don't need to stick to what you thought was good when you were 12. Most people in developed countries do just fine without math and science.

 

[ParticleGrl]

How can you stand to wade through the boring bits?

So part of the problem here might be that you don't actually like physics, or haven't found what you like. I legitimately enjoyed most of my physics courses, and most of the topics. While there were the occasional excruciatingly boring bits (crystallographic point groups come to mind), they were fairly rare.

If you find so much of physics utterly boring, what exactly is it you are interested in?

How do you ever GET to the good bits?

Have you tried simply grabbing an intro quantum field theory text and diving in? Don't worry about your diff-eq or group theory being rusty, you'll pick it back up and cement what you need while fighting through the "good bits." Knowing some of the physics can help focus your math study.

Start by figuring out the "good bit" you want to know, and then use that to focus your study.

How do you ever get to USE any of this material?

I used it for about 4 years while working on my phd thesis and have literally never had an opportunity to use any physics knowledge ever again.

I know my fate is most probably to be a dabbler for the rest of my life.

I went all the way through a phd and never held a job in physics after my phd program. Most of my cohort from grad school never held a job in physics after their phd. If you are making a good living tutoring physics, you actually probably use more physics knowledge in your day-to-day work than the majority of physics phds, so I wouldn't worry too much about dabbling.