Thank you for your comment yicong2011.
I've noticed a great theme in these Academic forums: people posting advice for good books to read to fill in knowledge gaps. Outside of what you can pull out of the internet jungle, there is a large, confused book market for postgraduate studies, e.g., the very expensive Springer books.
You can certainly learn mathematics as you study physics. I did this through my BS, and even through my doctoral work, despite having earned an MS in mathematics in between. However, like many others, I felt that much of what I had learned was simple aping of mathematical prescriptions to treat advanced physics problems. If your bent is for experimental physics, then you're probably okay with the mathematics you've pick up from school.
If you're into theory, unless one gets very lucky with a great advisor, one cannot pick up a book or research paper from the current, even old, literature and just read it. Straight away, you will get lost in nomenclature, forget about substance. It will take you years, if you have the time and the persistence, to succeed in understanding leading theoretical physics because there is a lot of poorly written junk out there, and because most of the literature out there is not meant to be pedagogical.
I recall the news splash a few years ago about a surfer dude (Lissi) having potentially unified the known forces. His paper was steeped in E8 group theory. Your not going to get knowledge of E8 from any undergrad or grad class I know of. So you google for articles and search Amazon for books. Work hard enough and long enough, and you will succeed in reading Lissi, or other papers in this area of physics steeped in group theory. It was a decade for me past my PhD, and I'm not a slouch, to be able to "catch up" and really read present day physics literature. While working at accelerators, or designing nukes at LANL, most of my time was wasted on trying to get something out some book or paper. I succeeded little by little, wasting a lot of time on searches on the internet, perusing new books at the LANL library afterhours, and spending more money for the next book sold on Amazon. The price per bit of knowledge gained was steep. Many times after long battles I have thought to myself, well, I've learned this much, and I'm just not smart enough to go to the next level, "the end", only to stumble by happenstance into the book or article that finally clicked for me.
Pick up the pedagogical book on String Theory or the new one in Loop Quantum Gravity. You'll likely get the big idea of strings, but I think the average reader will have far less success in LQG without reading Nakahara. And then you'll run into all of that algebraic topology in both the string theory and LQG books. Do you pay lip service? Then what can you really do in any knowledgeable way?
Knowledge is power. The physicist Eugene Wigner was hated by many American physicists, e.g., Slater, for developing the gruppenpest (group theory). The gruppenpest methodology is now a critical element of the Standard Model, and models beyond it, e.g., SUSY. Slater and his ilk just couldn't handle the math. They lost.
The books I list in the syllabus are well written and very pedagogical in nature. When I would finally crack idea X or method Y, I'd pass it on to my students over the years. I finally got to the point where to a make a list or a syllabus was natural. The material in the list is, honestly, what worked best for me. Individual chunks have also worked for many others out there as I can see in these forums when someone asks for a good starter book in, say, QED. What's in my "list" for QED matches well with the recommendations in the posts of other people. I should add that most of the material is timeless. One will not be replacing the content in Goldstein's Classical Mechanics, or Jackson's Electrodynamics very soon. Nor, for that matter, the group theoretic content of Gilmore or Jones, or the algebraic topology in Nakahara. Should there be a course for beginning graduate students in mathematics and physics covering the syllabus material?
The first four chapters of R. Gilmore '74, followed by the first four chapters of H. F. Jones, 2nd ed., followed by R. Gilmore 2008 will make your probability of understanding the work of Georgi (and his book) much more likely. They will also increase your understanding of mathematical physics to ordinary, graduate school core classes, far beyond aping prescriptions you copied from a chalkboard. Jackson problems will be clothed in a unified picture thanks to the work started by Sophus Lee.
I found my understanding of gauge field theory much improved by my reading of Nakahara and seeing gauge fields in a much more general, and mature language. The physics texts by Ryder, 1st or 2nd ed., only give you a hint of the deeper picture. If you are searching for new gauge field theories without all of the mathematics tools, it is very likey your work will be trivially dismissed by those who really know their stuff. There are very few geniuses out there, and the low hanging fruit is gone. I also find, knowing many doctoral level mathematicians at Los Alamos National Laboratory and beyond, that these mathematicians are hindered by not knowing the (typically physics based) history of their knowledge, nor knowing the applications to physical problems. Both camps lose.
There was an ariticle released not too long ago that, circa today (2011), physicists peak at 48, primarily because it takes so long to master the skill set, this being mostly advanced mathematics. My syllabus is intended to reduce this time by providing a minimal but sufficient set of material that can be self-learned in a relatively linear way. And pretty much every one of the books or research articles have tons of citations should one of the listed books or articles not resonate with you. The syllabus also provides you a history of how, say, classical mechanics has co-evolved over centuries with variational methods to give us the modern basis of quantum mechanics and quantum field theories. History helps, if only so that you avoid repeating it, unkowingly recreating likely dead old ideas. Personally, I find knowing the history of today's physics and mathematics more useful than this. I see connections between disparate things like Galois theory and the solution of partial differential equations. Knowing disparate connections is, I believe, important for making advancements in both mathematics and physics.
Of course, we're all entitled to our opinions, and we can do with our time as we see fit.
Cheers,
Alex Alaniz
