Here is a running bibliography (or rather, a list of stuff to read if you want to know something about what I do). Here I assume that the standard undergraduate stuff in both math and physics has been done (although I have a lot of opinions about THAT, too).
This list gives my perspective as a YOUNG researcher. That means that I'm a beginner myself, hence I freshly remember what material helped and what didn't. Naturally it is not possible to give an objective opinion about a book ONCE one already knows a subject - it all looks familiar and nice. Thus I will try to indicate what books I actually USED and what books I found later and seem nice. I also will try to indicate standard books that I DID NOT LIKE (so that the reader may avoid pitfalls, or at least have somebody who feels their pain). If you are the author of one of the books that I do NOT recommend then please don't take offense.
Update: I recently (March 2007) found that John Baez has a similar page with many references listed also. Check it out.
Basic Mathematics Tools
The following gives the BASIC set of math tools that any beginning grad student (mathematics and theoretical physics) should learn.
Differential Topology and (later) Lie Groups
Differential topology is where you will learn about manifolds (physics application = general relativity). Representations of Lie groups are used in many applications (such as quantum field theory). Some mathy books are:
For representation theory I recommend:
Functional analysis (aka infinite dimensional linear algebra) is used in classical mechanics, quantum mechanics, classical field theory, and quantum field theory. It is also everywhere in PDEs, applied math, math finance, etc. You'll learn about words like Banach Space, Hilbert Space, Fourier Transform, Distribution, Sobolev Space, Calculus of Variations, etc. A good book that I used is:
Used everywhere. You'll learn about groups, rings, and fields, and you'll lay a foundation for things like category theory (if you are interested in that)
Real and Complex Analysis
Used everywhere. You will learn about measure theory and Lebesgue integration (used in probability a lot), as well as about complex structures
More advanced students in geometry (especially algebraic geometry) will appreciate learning about complex geometry from these books (especially the first and second books!):
Not to be confused with ALGEBRAIC GEOMETRY!!! Not immediately useful to most physicists, but it certainly comes up often for mathematical physicists. You'll learn about words like fundamental group, homotopy groups, homology, and cohomology
Harder (but still MUST-KNOW) Math Tools
Here are the tools that you will need to learn after you have mastered the above basic math tools (so probably in your second/third years). Note that you might consider Lie Groups in this category as well, even though I grouped it with Differential Topology above.
Differential (Riemannian) Geometry
You will learn how to throw metrics (lengths and distances) on your manifolds. Applications are obvious in general relativity. Mathy books are:
The above books cover "affine connections". There is a different approach to diff. geometry called Cartan formalism (emphasizes "moving frames"). More generally, there is an approach based on principal bundles and Ehresmann connections. I recommend highly Spivak Vol II (mentioned above) for both of these topics. Here are some more resources:
Physicists learning conformal field theory will find this topic useful. Beginners and physicists can learn a lot from this lovely little book
Basic Physics Tools
The following should be done before/during the first year of graduate school. Unfortunately most physics departments take an approach that (in my opinion) is outdated and, frankly, not worthwhile. For example, FORGET Jackson's electromagnetism book unless you plan on going into something more applied (like plasma physics). Do you REALLY need to spend months learning how to do wave guide calculations? The basic treatment of Maxwell's Equations from undergrad should be a good place to start learning a much nicer/more geometric way to formulate these things (see, e.g., Baez's book discussed below).
Also, beware of graduate Statistical Mechanics/Thermodynamics classes. Perhaps I've had too many bad experiences (or too many bad books).
Classical Mechanics is also a very sore spot for me. Instead of focusing on what's important (like constrained systems, etc) most professors of mechanics come from Nonlinear Dynamics - a worthy field, but (like Stat Mech) typically a zoological study of different models (most of which probably won't apply to YOUR research, and should be really left for those who are going into those fields). Speaking of classical mechanics...
If you are interested in learning about the foundations of quantum mechanics then you need to first learn about how to generalize classical mechanics.
You can start with these standards:
Jose and Saletan (seems good, but I cannot be objective because I found it too late) Goldstein (NOT my favorite)(Again I recommend taking in moderation all of the applications to nonlinear dynamics - out of these textbooks you need to know the basics of Lagrangian and Hamiltonian Mechanics - then MOVE on). Getting more advanced you can TRY:
Abraham and Marsden (NOT my favorite) Arnold (NOT my favorite)The notation is a little funky - I definitely think that there is plenty of room for a decent book about classical mechanics to be written.
Probably the cleanest 10 pages that I ever read on the subject was the first section of
Mackey, Mathematical Foundations of Quantum MechanicsThe second half is awesome as well (if you care about the mathematical foundations of quantum mechanics). Unfortunately the age of the book becomes evident because his manifold notation is arcane. I was pretty happy reading the first section or two and then generalizing it to arbitrary manifolds myself. Still, those 1950's masters were very clear thinkers.
Note that you will need to know something about CONSTRAINED Hamiltonian systems if you ever want to understand anything about gauge theory or really anything beyond the simplest quantum systems. For this I recommend the first several chapters from:
Henneaux and Teitelboim, Quantization of Gauge Systems Dirac, Lectures on Quantum MechanicsObviously the rest of these books will come in handy later for the quantum counterpart of this constrained story.
There are countless books devoted to basic quantum mechanics. Standards are (and they're fine I think)
Griffiths (nice beginner undergrad text) Sakurai (nice first year of grad school)When you get serious about the mathematical FOUNDATIONS of QM then I recommend (don't start with these, though):
Dirac, Lectures on Quantum Mechanics Mackey, Mathematical Foundations of Quantum Mechanics Segal (Irving), Mathematical Problems of Relativistic Physics von Neumann, Mathematical Foundations of Quantum Mechanics Folland, Harmonic Analysis on Phase SpaceI know that some (most) of these are old... but the mathematical treatment there is still lovely (especially the middle three).
Note that I think that it is a SHAME that all major approaches to quantization are NOT taught in basic quantum mechanics (usually the operator/Hamiltonian formalism is described, but the path integral/Lagrangian quantization is left for quantum field theory - making both subjects unnecessarily more difficult to learn. Usually deformation quantization is ignored altogether; deformation quantization shows how the operator formalism can really be viewed merely as a deformation of the classical formalism).
Harder (but still mandatory) Physics
The following lists the first/second year material in physics that is more difficult, but definitely "BASIC" for a theorist:
Classical Field Theory
Just as you want to talk about classical mechanics BEFORE quantum mechanics, you also want to discuss CLASSICAL FIELD THEORY before Quantum Field Theory. Unfortunately, this just ISN'T DONE. Most departments give you a class in Jackson's electromagnetism and call it good (don't get me started). This stuff seems to be the sort of thing that you must learn from somebody else, probably LONG after you think that you know something about QFT (you'll probably be deep in the details of Feynman diagrams before you realize that you don't know squat about even classical field theory). A SHAME SHAME SHAME!!! Most QFT books give a "Chapter 1" ripoff treatment.
Personally I'm in love with anything that John Baez writes:
Baez and Muniain, Gauge Fields, Knots and GravityI can recommend this book heartily - it will definitely help (be careful: Baez has a conference proceedings of similar name - you probably don't want that).
Quantum Field Theory
Jumping on to quantum field theory you will find a variety of texts. Since there are a couple of ways of quantizing some books are better for some things than others. I first recommend:
Ryder, Quantum Field Theory (good for path integral/Lagrangian formulation) Lahiri and Pal (good for canonical/Hamiltonian formulation) Michael Stone, The Physics of Quantum Fields (condensed matter approach, which is more motivated than high energy) Zee (very friendly discussion - I wish that I had read it when I was first starting)
Books like Peskin and Schroeder (encyclopedic) or Bjorken and Drell (older) seem really better as REFERENCE than as first reads. Peskin and Schroeder run headfirst into the nuts and bolts of particle physics, hence I found it hard to discern the difference between fundamental QFT concepts versus nitty-gritty details of the Standard Model (when I was first starting). It IS a useful reference, though. I have limited experience with Weinberg's books.
When you get more advanced then there are books like
Haag, Local Quantum Physics: Fields, Particles, Algebras Coleman, Aspects of Symmetry Freed, Five Lectures on Supersymmetry Freed et al, Quantum Fields and Strings: A Course for Mathematicians (two volume beast)
Haag's book is based around an approach to QFT called "Algebraic QFT" which relies heavily on C*-algebras (and more particularly von Neumann algebras) - see below for references concerning those subjects. Freed's books demand a very mature perspective. Most things I didn't understand until I had extended conversations with him :). Actually, most things I STILL don't understand.
Before you run off and learn General Relativity you SHOULD learn some basic differential geometry (which, in turn, means that you should learn some basic differential topology - see above). If you are impatient then you can go on (I guess), but you'll just have to come back and clean up later. Here are some good ones listed in order of difficulty (there are MANY more)
D'Inverno, Introducing Einstein's Relativity (my friends love it) Sean Carroll, notes available online (also a printed version) (VERY FRIENDLY BOOK for beginners) Wald, General Relativity Misner, Thorne, Wheeler - never read it myself... the big black bible
Topological Quantum Field Theory
When I have time some day then I'll actually do this paragraph some justice. Witten's original paper is still probably the best start:
Witten, Quantum Field Theory and the Jones Polynomial, Comm Math Phys 121, 351-399 (1989)Atiyah wrote a nice little book called
Atiyah, Geometry and Physics of Knots (1990)I have MUCH more to say about this, but that will have to wait.
Quantum Groups/Modular Tensor Categories
Quantum groups/Modular Tensor Categories comprise the algebraic and/or categorical approach to the topological quantum field theories discussed above. Here are three excellent books about MTCs (I have much more to say, but later)
Turaev, V., Quantum Invariants of Knots and 3-Manifolds Kassel, Quantum Groups Bakalov and Kirillov, Lectures on Tensor Categories and Modular FunctorThe last book is still available ONLINE!!! Check out their website and follow the links there for more material.
People didn't come up with MTCs out of thin air. For example, the original papers of Reshetikhin and Turaev deal with certain types of quantum groups. These are examples of MTCs.
Here are other useful books for reference:
Prasolov and Sossinsky, Knots, Links, Braids and 3-Manifolds Rolfsen, Knots and Links Gompf and Stipsicz, 4-Manifolds and Kirby Calculus
Loop Quantum Gravity (I'm not an LQG researcher)
If you want to start learning about Loop Quantum Gravity then I would definitely start with John Baez's stuff (read anything by Baez - he's very COOL). A good intro which has topics ranging from basic Riemannian geometry up through knot theory, Yang Mills theory, Chern Simons theory, and Ashtekar's new variables for gravity is
String Theory (I'm not a string theorist)
If you want to start learning String Theory then I recommend starting with
As far as learning CFT is concerned, I'm still searching for the best references. Here are the usual references:
This hobby has brought me a lot of joy. If you want to learn about deformation quantization then it is probably best to start with some of the easier introductions (on the ArXiv) (note: you maybe first want to learn the "density operator" formalism of quantum mechanics, which is a more general way to characterize "state". Try the standard quantum computation books since they use that heavily):
Cosmos Zachos, Deformation Quantization: Quantum Mechanics Lives and Works in Phase Space Hirshfeld and Henselder, Deformation Quantization in the Teaching of Quantum MechanicsThese papers will give you an easy idea about deformations of the observable algebra and how so-called Wigner functions play the role of "states" (although the notion of "state" is more general than Wigner functions - see e.g. Waldmann's papers on the ArXiv).
It really pays to get the idea behind some of the early work due to Wigner and Moyal. Some of this stuff is great just for addressing the foundations of quantum mechanics in general. I recommend
H.W. Lee, Physics Reports 259 (1995) pg 147-211 Wigner et al, Physics Reports 106 (1984) pg 121 Groenewold, On the Principles of Elementary Quantum Mechanics, Physica 12, 405-460 (1946) Agarwal and Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics I, Phys. Rev. D10, 2161-2186 (1970)
The modern viewpoint had its genesis in these papers:
Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer, Annals of Physics 111, (1978) 61-110 Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer, Annals of Physics 110, (1978) 111-151
After that def. quant. matured under the works of Fedosov, Kontsevich, and many others. Check out the deformation quantization homepage for references.
C*-algebras give an interesting perspective on quantization since the operator algebra has this structure (gives a different view on "operators on a Hilbert space"). Furthermore, it is possible to actually CONSTRUCT "states" as a derived concept. In essence, it is possible to BUILD the Hilbert space rather than just postulate it.
If the algebra is commutative (classical mechanics) then the state space is constructed using the Gelfand-Naimark theorems. If the algebra is noncommutative (quantum mechanics) then the appropriate tool is the GNS construction (Gelfand-Naimark-Segal).
C*-algebras are standard material: the following textbooks have served me well
Murphy, C*-Algebras and Operator Theory Fillmore, A User's Guide to Operator Algebras Davidson, C*-Algebras by ExampleC*-algebras are also necessary if you are interested in studying Noncommutative Geometry (Connes - see next).
The are several flavors of "noncommutative geometry" - in this paragraph I mean the work of Connes et al which is based on a generalization of the notion of "space" by building it using C*-algebras. This is VERY MUCH in the spirit of algebraic geometry since somehow we are characterizing a space by some functions on this space (the functions form a C*-algebra, but here the functions don't have a commutative pointwise product). I recommend the following:
SEE THE C* books ABOVE FIRST!!! Landi, An Introduction to Noncommutative Spaces and their Geometry, hep-th/9701078 (See the references in there to Kaster's reviews as well as the review by Gracia-Bondia and Varilly) Connes, Noncommutative Geometry (I DON'T understand this book)
Other random books that I like
Milnor and Husemoller, Symmetric Bilinear Forms Conway and Sloane, Sphere Packings, Lattices, and Groups Milnor and Stasheff, Characteristic Classes Grimmett and Stirzaker, Probability and Random Processes
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