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. Just think of me as a fool :)
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 of mathematics or theoretical physics should know. Without these
tools I just don't know how to communicate effectively.
Differential Topology and (later) Lie Groups
Both are standard subjects that ANYBODY needs to do anything, e.g.
Lie groups are needed in QFT, differential topology is needed for
differential geometry (which, in turn, is needed for general relativity), etc.
Personally, I don't know how most physicists learn GR without knowing this
material first. The standard books are
Guillemin and Pollack, Differential Topology Warner, Foundations of Differentiable Manifolds and Lie Groups Spivak, Calculus on Manifolds (I own a copy, but I never read it) Spivak, A Comprehensive Introduction to Differential Geometry Vol I (and II is useful, too).Many physics graduate students tend to prefer the following book (which looks friendly, but I have little personal experience with it):
Schutz, Geometrical Methods in Mathematical PhysicsI'm being a little cheap with the Lie Groups references, I admit. Spivak (Comprehensive series) also covers most of basic DIFFERENTIAL GEOMETRY (see below) - I highly recommend those two books.
Most of the interesting aspects of Lie groups involve representation theory. For this I highly recommend starting with:
Fulton and Harris, Representation Theory: A First Course Brian Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Functional Analysis
You'll need a lot of functional analysis (aka infinite dimensional linear algebra)
to understand anything in: classical/quantum
mechanics/field theory, C*-algebras, or really anything. You'll learn about words
like Topological Vector Space, Banach Space, Hilbert Space, Fourier Transform,
Distribution, Sobolev Space, Calculus of Variations, etc). A good book that
I used is:
Reed and Simon, Functional AnalysisMany people have told me that a good book (especially for first-timers) is Kreyszig's book
Kreyszig, Introductory Functional Analysis with Applications
Algebra
This language will live with you forever, so you'd better get good. It's pretty
standard. I learned from (a very good, but big, book)
Dummit and Foote, Abstract AlgebraPersonally I didn't like Lang's book because his proofs are based on a more categorical approach, which is no way to START the subject (although maybe that's useful later - I've never bothered to look again). Isaacs' book was FAR too expensive - what's his publisher's problem? Don't support that kind of crap - burn Brooks Cole Publishing.
Real and Complex Analysis
There's no way out... if you don't then you won't amount to squat. For Real
Analysis use
Rudin, Real and Complex Analysis (green book) Wheeden and Zygmund, Measure and Integral (quite nice) Folland, Real AnalysisBEWARE: Rudin has a blue (undergrad) book - I've never looked at it. For 1-variable complex analysis I started with
Gamelin, Complex AnalysisI liked his "breezy" style - makes learning easy.
More advanced students of geometry (especially algebraic geometry) will appreciate learning about complex geometry from these books (especially the first and second books!):
Griffiths and Harris, Principles of Algebraic Geometry Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves, and Moduli Spaces (beginners and physicists will love this book) Birkenhake and Lange, Complex Abelian Varieties (more advanced)
Algebraic Topology
Not to be confused with ALGEBRAIC GEOMETRY!!!
This is also a first-year
course, and not immediately of use to your typical physicist, but it
certainly comes up for mathematical physicists, so it is mandatory.
Here you'll learn about words like fundamental group (and higher
homotopy groups), covering spaces, homology, and cohomology:
Armstrong, Basic Topology (undergraduate text) Massey, Algebraic Topology: An Introduction (still pretty easy) Hatcher, Algebraic Topology (excellent, and available online)All 3 books are useful. Some of you might skip directly to Hatcher.
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
It is
impossible to do physics without this - so it should be learned WELL. I used
Spivak, A Comprehensive Introduction to Differential Geometry Vol I (and II is very useful). Do Carmo, Riemannian Geometry (decent) Petersen, Riemannian Geometry (a bit too formal for my taste)Again, physics students might want to look at Schutz's book (mentioned in the differential topology section above).
The above books cover "affine connections". There is a different (but very beautiful) 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:
Kobayashi and Nomizu, Foundations of Differential Geometry Vol I and IIOnce you know this, General Relativity will come much easier (although there they use Lorentzian rather than Riemannian signature).
Algebraic Geometry
Still scares the hell out of me.
At the risk of making a fool out of myself, I don't care about
books that are too formal (except for reference), and I don't
care about proofs (at least
the first time through a subject). Intuition and
calculation technique are more important. Beginners and
physicsts can learn a LOT from this lovely little book
Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves, and Moduli Spaces
Here is a standard book that I highly recommend:
Griffiths and Harris, Principles of Algebraic GeometryThose who come from a more algebraic bent (i.e. those raised in unloving homes) will want to work with these:
Hartshorne, Algebraic Geometry Mumford, Red Book of Varieties and Schemes (there MAY be an illegal online copy somewhere)Definitely a second/third year graduate course - you might need some extra Commutative Algebra as well.
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 some nasty 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, but typically these courses involve months of zoological study of the phase transition properties of many many systems. I'm not sure what tools/techniques/knowledge I walked away with. Maybe you'll fare better. Perhaps try conformal field theory (see below) or statistical field theory.
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...
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.
Quantum Mechanics
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) 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.
General Relativity
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 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
RESEARCH AREAS
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 two excellent books
about MTCs (I have much more to say, but later)
Turaev, V., Quantum Invariants of Knots and 3-Manifolds Bakalov and Kirillov, Lectures on Tensor Categories and Modular FunctorThe latter 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". The representation categories of these quantum groups happen to be examples of MTCs.
There's a lot to say about quantum groups. Since I'm out of time today, here's a standard reference.
Kassel, Quantum Groups
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
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
Baez and Muniain, Gauge Fields, Knots and Gravity
I haven't read Pullin's book, but everybody gives it high acclaim. From here you seem to be left to the papers - mostly available on www.arxiv.org. So far I like:
Ashtekar and Lewandowski, Background Independent Quantum Gravity: A Status Report, gr-qc/0404018 Thiemann, gr-qc/0210094 (notes) Thiemann, gr-qc/0110034 (book)Personally I found Thiemann's stuff "technical-heavy", but relevant. Spin foams are all the rage these days:
Perez, Introduction to Loop Quantum Gravity and Spin Foams, gr-qc/0409061 Perez, Spin Foam Models for Quantum Gravity, gr-qc/0301113
String Theory (although I'm not a string theorist)
If you want to start learning String Theory then I recommend browsing FIRST through
Zweibach, A First Course in String Theoryit's EASY and gives you a MOTIVATED introduction to String Theory. Undergrads can understand it (I think). It's probably overkill for a grad student to read it in detail and work the problems (think of it as weekend reading). Most people will guide you to
Green, Schwartz, and Witten, Superstring Theory (Vol I and II) (old approach) Polchinski, Introduction to String Theory (Vol I and II) (new approach)Green, Schwartz, and Witten is clearer AFTER looking at Zweibach. I cannot say that I have learned much from looking at Polchinski. He basically throws conformal field theory (CFT) and the operator product expansion (OPE) at you with little/no motivation. Go learn CFT elsewhere first.
As far as learning CFT is concerned, I'm still searching for the best references. Here are the canonical references:
Di Francesco, Mathieu, Senechal, Conformal Field Theory (encyclopedic) Ginsparg, hep-th/9108028 (starts fine, but loses motivation quickly)The best references that I use (but I'm a geometer) are
Toshitake Kohno, Conformal Field Theory and Topology (AMS 2002) Graeme Segal, Definition of Conformal Field Theory (appeared in the book Topology, Geometry and Quantum Field Theory, Cambridge Univ. Press 2004)
Quantum Computation
I can only give advice on the material that I've read. There is
standard material out there that I haven't looked at, but for now
I recommend heartily
Kitaev, Shen, Vyalyi, Classical and Quantum ComputationJohn Preskill has a set of notes that I read a million years ago - they're useful but rather dated - hence I recommend the more modern textbooks.
Deformation Quantization
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 should first 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 in def quant is still not well-developed - see 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 great works of Fedosov, Kontsevich, and many others. Check out the deformation quantization homepage for references.
C*-algebras
C*-algebras are an integral part of quantization since the operator algebra
has this structure (this is more general than just "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 this falls under 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 (a la Connes).
Noncommutative Geometry
The are several flavors of "noncommutative geometry" - in this paragraph I mean
the work of Connes et. al. which is based around some generalization
of the notion of "space" by building it using C*-algebras. This is
still 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 have looked at the following:
SEE THE C* books ABOVE!!! Connes, Noncommutative Geometry (I DON'T understand this book) Landi, An Introduction to Noncommutative Spaces and their Geometry (available on ArXiv)
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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