The study of particles and fields is not made easier by all of the seemingly disparate physics ideas and mathematical methods. I put together an 8 page syllabus of math and physics books, as well as reference literature, to take you from a junior level math or physics background to graduate/post-graduate QFT, step by step. My goal was singular: pull out the minimal set of math and physics ideas, backed up by the actual history, that underlie current methods of theoretical physics, and find the best, clearest books and literature to learn them--about a decade of work after finishing a doctoral degree in physics. To this short syllabus I added a 50 page document with more math/physics history underlying physics foundations, including the limits in math foundations and the limits inherent in physics that we've run into. The longer document trys to give you the historical basis to the big ideas you shouldn't miss in Calculus I, II, III, linear algebra, modern algebra, topology,...,classical mechanics and electrodynamics, quantum physics, QED,... Example books in the short syllabus nearer the beginning sections: 1. Mathematics: Calculus of variations a. Calculus of Variations, L. D. Elsgolc, Dover Publications. Originally written in Russian, this book was first published in English in 1961. Using clear notation, Elsgolc develops the calculus of variations side-by-side with ordinary differential calculus. Starting with a challenge to Isaac Newton, this calculus originated from extremization problems in physics, e.g., least time, maximum entropy, least action. The Standard Model, general relativity, string theories, to name but a few, are expressible in terms of least action. Ideally this book should be read before graduate work in physics, probably concurrently with junior level mechanics. b. Variational Principles in Dynamics and Quantum Theory, W. Yourgrau and S. Mandelstam, Dover Publications. Tracing the evolution of the concept of the innate economy of nature (least action) from the Greeks through to Fermat’s principle of least time and Maupertuis’ le principe de la moindre quantité d’action (least action) in 1744, this book traces the development of the equations of Lagrange, Hamilton, Hamilton-Jacobi, etc., in classical mechanics and electrodynamics to the various historical paths to quantum physics including those of Feynman and Schwinger. This book should probably be read concurrently during the first year of graduate school, if not at the completion of the undergraduate degree. Without these readings, or similar, the use of the principle of least action is little more than a physics gimmick. . . . Examples books nearer the end sections of the short syllabus: . . . 2. Quantum Electrodynamics, W Greiner and J Reinhardt, 3rd ed., Springer. Another pedagogical text, this book presents, detail by detail, QED the old fashioned way, the way people, including Feynman, first developed QED. I believe this is an essential read if you want to understand QFT, including some of the early issues with divergences. 3. Quantum Field Theory, 2nd ed., L. H. Ryder—This is a very well written introductory text on QFT up through introductory supersymmetry (SUSY). Ryder surveys relativistic wave equations and Lagrangian methods, the quantum theory of scalar and spinor fields, and then the guage fields. In chapter 3, Ryder carefully explains the principle of minimal coupling by requiring local invariance of, firstly, the Lagrangian for the complex scalar field before moving on to treat Yang-Mills fields. Also in chapter 3, Ryder points out the parallels originating from parallel transport in general relativity in the framework of a manifold, and parallel transport in the algebra of quantum fields in the framework of continuous Lie groups. Is this parallel an accident? Is there a deeper level of grammar? The answers to these questions are not easily found in popular QFT texts, nor in books on the mathematics of differential geometry, group theory, or algebraic topology. In fact, most books purportedly treating these areas of mathematics for the physicist also fail to deliver the deeper grammar. 4. Lie Groups, Lie Algebras, and Some of Their Applications, R. Gilmore, Dover, was originally published in 1974. In the same sense that the two books on the calculus of variations, Elsgolc 1961, and W. Yourgrau and S. Mandelstam 1968 provide the fundamental least action underpinnings of classical and quantum physics, Gilmore provides the foundations to the prescriptions in standard QFT books. The physicist’s book, “Lie Algebras In Particle Physics, From Isospin to Unified Theories,” 2nd ed., H. Georgi makes a ton more sense after Gilmore. If I had stumbled across Gilmore sooner, I probably wouldn’t have spent years pouring over a ton of pure mathematics texts, never quite understanding how to bridge pure algebraic topology back to quantum fields. The following books should probably be read in reverse order from the way I found and read them. They are: 5. Groups, Representations And Physics, 2nd ed., H. F. Jones, Institute of Physics Publishing. This was the first book that took me a long way into both understanding and being able to apply group theoretic methods to quantum mechanics and quantum fields. After working through Jones, however, I still felt there was a deeper plane of truth, or a better grammar if you will. There was still too much “genius”, too much particularization. Before reading Jones, I recommend as a minimal prerequisite an introductory text on group theory at the Schaum’s outline level. I personally like, “Modern Algebra, An Introduction”, 2nd ed., J. R. Durbin, Wiley. You need only cover the material up through group theory. Take with you the notion of a normal subgroup when you proceed to read Gilmore. 6. Lie Algebras In Particle Physics, From Isospin to Unified Theories, 2nd. ed., H. Georgi, Frontiers in Physics. I couldn’t have read this book without first having read and worked through Jones. Georgi was difficult for me, but when I cracked it, I began to feel like I was starting to understand the physicist instead of the mathematician. Ideally, read the first four chapters of R. Gilmore’s text first. The 5th chapter covers applications to areas typically presented in graduate physics coursework. Then read Jones, then Georgi. There will be much less for you to have to accept by fiat. . . . At this point you now have a path to the underpinnings of two major chunks of the mature grammar of modern QFT theorizing: the principle of least action, and tools for studying the structure of Lie groups and Lie algebras (and particle spectra). Still missing is a deeper, more general understanding of the principle of minimal coupling resulting from the requirement of local invariance of Yang-Mills Lagrangian densities. To maintain invariance under local transformations—the steps being presented very clearly in Ryder—requires the addition of extra terms. In the case of the complex scalar field, the extra terms correspond to electrodynamics expressed in the potential formulation—hence why I suggest you understand electrodynamics at least at the level of Griffiths. Your radar should be on, looking for a more mature grammar to the principle of minimal coupling, especially after reviewing basic Kaluza-Klein theory. Kaluza-Klein theory was an early attempt to unify electromagnetism with gravity, and it goes like this: An extra fifth spatial dimension can be understood to be the circle group U(1) as electromagnetism. Electromagnetism can be formulated as a guage theory on a fiber bundle, namely the circle bundle with guage group U(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group. Such generalizations are often called Yang–Mills theories in flat spacetime, as opposed to curved spacetime in Kaluza-Klein theory. Note that Kaluza-Klein theory (in any (pseudo-)Riemannian manifold, even a supersymmetric manifold) can be generalized beyond 4 spatial dimensions. So is there a good book out there to view this approach in a more unified way? Yes. 8. Geometry, Topology and Physics, M. Nakahara, Graduate Student Series in Physics, chapter 9. I read the first four chapters before skipping to chapter 9. To me, chapter 9 seems fairly self-contained. However, by the time I happened upon Nakahara, my background in mathematics was far beyond my 36 hour masters degree in pure mathematics, and I also knew what the goal was beforehand thanks to another book, namely, “Topology, Geometry, and Gauge Fields, Foundations,” G. L. Naber. Naber sucks. Naber is part of the reason I overdid mathematics, but Naber put the goal, the mature grammar in easy to understand words. “…These Lie algebra-valued 1-forms…are called connections on the bundle (or, in the physics literature, guage potentials).” The guage fields in QFTs are connections over principle bundles. If anything, you have to read Naber’s chapter 0 for motivation, and I’ve reluctantly come to appreciate all of the mathematics I studied trying to get through Naber, especially differential forms. At this point I began to see that there is probably no end to physics theoreticians cooking up hypothetical universes that don’t necessarily have to have anything to do with what we perceive to be our universe. Even theorizing over our own apparent universe is probably unlimited. The creative degrees of freedom to cook up mathematical universes that behave at low energy like what we observe seem infinite. As our experimental knowledge grows, we exile certain theories of physics into the realm of mathematics, only to quickly create a whole new frontier of endless physics-based possible universes. This realization took the wind out of my pursuing my belief in Einstein’s dream of a final theory. By the way, I found a pretty tidy review of differential forms online, namely, “Introduction to differential forms,” D. Arapua, 2009. I was never satisfied by any of the physics books purportedly written to teach forms. You may have to find books/literature that make more sense to you, but at least, hopefully, you should have the historical paths to current ideas. Cheers, Alex PS--Can't upload the long document at 248 KB. Ping me for a copy.