books
How to Learn Math and Physics
John Baez
October 14, 2024
Introduction
"How to learn math and physics" — the title is deliberately provocative.
Everyone has to learn their own way. I don't know
how you should learn math and physics. But presumably
you came here looking for advice, so I'll give you some.
My advice is aimed at people who are interested in fundamental
theoretical physics and the math that goes along with that. (By
"fundamental" physics I mean the search for the basic laws
concerning matter and the forces of nature.) If you want to do
experiments instead of theory, or other of kinds physics like condensed
matter physics and astrophysics, or math that has nothing to do with
physics, my advice will be of limited use. You should still learn the
basics I mention here, but after that you'll have to look
elsewhere for suggestions.
Learning math and physics takes a whole lifetime. Luckily, it's a lot
of fun... if you have a reasonably patient attitude. A lot
of people read pop books about quantum mechanics, black holes, or
Gödel's theorem, and immediately want to study those subjects.
Without the necessary background, they soon become frustrated — or
worse, flaky.
It can be even more dangerous if you want to plunge into
grand unified theories, or superstrings, or M-theory.
Nobody knows if these theories are true! And it's hard to
evaluate their claims until you know what people do know.
So, especially when it comes to physics, I urge you to start with
slightly less glamorous stuff that we know to be true — at
least as a useful approximation, that is — and then, with a solid
background, gradually work your way up to the frontiers of knowledge.
Even if you give up at some point, you'll have learned something
worthwhile.
This webpage doesn't have lots of links to websites. Websites just
don't have the sort of in-depth material you need to learn technical
subjects like advanced math and physics — at least, not yet. To learn
this stuff, you need to read lots of books. I will list some
of my favorites below, and also some you can get free online.
But, you can't learn math and physics just by reading books! You have
to do lots of calculations yourself — or experiments, if you want to
do experimental physics. Textbooks are full of homework problems, and
it's good to do these. It's also important to make up your own
research topics and work on those.
If you can afford it, there's really nothing better than taking
courses in math and physics. The advantage of courses
is that you get to hear lectures, meet students and professors,
and do some things you otherwise wouldn't — like work your
butt off.
It's also crucial to ask people questions and explain
things to people — both of these are great ways to learn stuff.
Nothing beats sitting in a cafe with a friend, notebooks open, and
working together on a regular basis. Two minds are more than
twice as good as one!
But if you can't find a friend in your town, there are different ways
to talk to people online. In all cases, it's good to spend some time
quietly getting to know the local customs before plunging in and
talking. For example, trying to start a rambling discussion on a
question-and-answer website is no good. If you've got physics questions,
try Physics Stack
Exchange. For research-level questions, try Physics Overflow. For questions
about math, try Math Stack
Exchange, or for research-level questions, Math Overflow.
For more free-wheeling
discussions of math and physics, try Physics Forums.
There are also lots of interesting blogs and free
math books online.
Finally, it's crucial to admit you're wrong when you screw up.
We all make tons of mistakes when we're learning stuff. If you don't
admit this, you will gradually turn into a
crackpot who clutches on to a stupid
theory even when everyone else in the world can see that it's wrong.
It's a tragic fate, because you can't even see it's happening.
Even bigshot professors at good universities can become crackpots once
they stop admitting their mistakes.
To avoid looking like a fool, it's really good to get into the habit
of making it clear whether you know something for sure, or are just
guessing. It's not so bad to be wrong if you said right from the
start that you weren't sure. But if you act confident and turn out to
be wrong, you look dumb.
In short: stay humble, keep studying, and you'll keep making
progress. Don't give up — the fun is in the process.
How to Learn Physics
There are 5 cornerstone topics that every physicist should learn:
and
in roughly that order. Once you know these, you have
the background to learn the two best theories we have:
and
And once you know these, you'll be ready to study
current attempts to unify quantum field theory and general relativity.
If this seems like a lot of work... well, it is! It's a lot of
fun, too, but it's bound to be tiring at times. So, it's also
good to read some histories of physics. They're a nice change of
pace, they're inspiring, and they can show you the "big
picture" that sometimes gets hidden behind the thicket of
equations. These are some of my favorite histories:
-
Emilio Segre,
From Falling Bodies to Radio Waves: Classical Physicists and Their
Discoveries, W. H. Freeman, New York, 1981.
-
Emilio Segre,
From X-Rays to Quarks: Modern Physicists and Their Discoveries,
W. H. Freeman, San Francisco, 1980.
-
Robert P. Crease and Charles C. Mann,
The Second Creation: Makers of the Revolution in Twentieth-Century
Physics, Rutgers University Press, New Brunswick, NJ, 1996.
-
Abraham Pais,
Inward Bound: of Matter and Forces in the Physical World,
Clarendon Press, New York, 1986. (More technical.)
Next, here are some good books to learn "the real stuff".
These aren't "easy" books, but they're my favorites.
First, some very good general textbooks:
- M. S. Longair, Theoretical Concepts in Physics,
Cambridge U. Press, Cambridge, 1986.
- Richard Feynman, Robert B. Leighton and Matthew Sands,
The Feynman Lectures on Physics, 3 volumes, Addison-Wesley, 1989.
All three volumes are now free online.
- Ian D. Lawrie, A Unified Grand Tour of Theoretical Physics,
Adam Hilger, Bristol, 1990.
Then, books that specialize on the 5 cornerstone topics I listed above:
Classical mechanics:
Statistical mechanics:
Electromagnetism:
Special relativity:
Quantum mechanics:
These should be supplemented by the general textbooks above,
which cover all these topics. In particular, Feynman's Lectures
on Physics are incredibly valuable.
After you know this stuff well, you're ready for general
relativity (which gets applied to cosmology) and quantum field theory
(which gets applied to particle physics).
General relativity — to get intuition
for the subject before tackling the details:
-
Kip S. Thorne,
Black Holes and Time Warps: Einstein's Outrageous Legacy,
W. W. Norton, New York, 1994.
-
Robert M. Wald,
Space, Time, and Gravity: the Theory of the Big Bang and Black Holes,
University of Chicago Press, Chicago, 1977.
-
Robert Geroch, General Relativity from A to B,
University of Chicago Press, Chicago, 1978.
General relativity — for when you get serious:
-
R. A. D'Inverno,
Introducing Einstein's Relativity,
Oxford University Press, Oxford, 1992.
-
J. B. Hartle,
Gravity: An Introduction to Einstein's General Relativity,
Addison-Wesley, New York, 2002.
-
B. F. Schutz,
A First Course in General Relativity,
Cambridge University Press, Cambridge, 1985.
General relativity — for when you get really serious:
-
Charles W. Misner, Kip S. Thorne and John Archibald Wheeler,
Gravitation, W. H. Freeman Press, San Francisco, 1973.
-
Robert M. Wald, General Relativity,
University of Chicago Press, Chicago,
1984.
-
Mattias Blau, Lecture Notes on General Relativity.
Cosmology:
-
Edward R. Harrison, Cosmology, the Science of the Universe, Cambridge
University Press, Cambridge, 1981.
-
M. Berry, Cosmology and Gravitation, Adam Hilger, Bristol, 1986.
-
John A. Peacock, Cosmological Physics, Cambridge
University Press, Cambridge, 1999. (More technical.)
Quantum field theory — to get intuition for the subject before
tackling the details:
Quantum field theory — for when you get serious:
-
Michael E. Peskin and Daniel V. Schroeder,
An Introduction to Quantum Field Theory,
Addison-Wesley, New York, 1995. (The best modern textbook, in
my opinion.)
-
Anthony Zee, Quantum Field Theory in a Nutshell, Princeton
University Press, Princeton, 2003. (Packed with wisdom told in a
charmingly informal manner; not the best way to learn how to calculate
stuff.)
-
Warren Siegel, Fields, available for free
on the arXiv.
-
Mark Srednicki, Quantum Field Theory, available free
on his
website.
(It's good to snag free textbooks while you can, if they're not on the
arXiv!)
-
Sidney Coleman, Physics 253: Quantum Field Theory,
1975-1976. (Not a book — it's a class! You can download free
videos of this course at Harvard, taught by a brash and witty
young genius.)
Quantum field theory — two classic older texts that cover a lot
of material not found in Peskin and Schroeder's streamlined
modern presentation:
-
James D. Bjorken and Sidney D. Drell, Relativistic Quantum Mechanics,
New York, McGraw-Hill, 1964.
-
James D. Bjorken and Sidney D. Drell, Relativistic Quantum Fields,
New York, McGraw-Hill, 1965.
Quantum field theory — for when you get really serious:
- Sidney Coleman, Aspects of Symmetry, Cambridge U. Press, 1989.
(A joy to read.)
- Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras,
Springer, 1992.
Quantum field theory — so even mathematicians can understand it:
-
Robin Ticciati, Quantum Field Theory for Mathematicians,
Cambridge University Press, Cambridge, 1999.
-
Richard Borcherds and Alex Barnard,
Lectures On
Quantum Field Theory.
Particle physics:
- Kerson Huang, Quarks, Leptons & Gauge Fields,
World Scientific, Singapore, 1982.
- L. B. Okun, Leptons and Quarks, translated from Russian by V. I. Kisin,
North-Holland, 1982. (Huang's book is better on mathematical aspects of
gauge theory and topology; Okun's book is better on what we actually
observe particles to do.)
- T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood,
1981.
-
K. Grotz and H. V. Klapdor, The Weak Interaction in Nuclear, Particle, and
Astrophysics, Hilger, Bristol, 1990.
While studying general relativity and quantum field theory,
you should take a break now and then and dip into this book: it's
a wonderful guided tour of the world of math and physics:
- Roger Penrose, The Road to Reality: A Complete Guide
to the Laws of the Universe, Knopf, New York, 2005.
And then, some books on more advanced topics...
The interpretation of quantum mechanics:
-
Roland Omnes,
Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994.
This is a reasonable treatment of an important but incredibly
controversial topic. Warning: there's no way to understand
the interpretation of quantum mechanics without also being able
to solve quantum mechanics problems — to understand the
theory, you need to be able to use it (and vice versa). If you don't
heed this advice, you'll fall prey to all sorts of nonsense that's
floating around out there.
The mathematical foundations of quantum physics:
-
Josef M. Jauch, Foundations of Quantum Mechanics,
Addison-Wesley, 1968. (Very thoughtful and literate. Get a taste of
quantum logic.)
-
George Mackey, The Mathematical Foundations of Quantum
Mechanics, Dover, New York, 1963. (Especially good for
mathematicians who only know a little physics.)
Loop quantum gravity and spin foams:
-
Carlo Rovelli, Quantum Gravity, Cambridge University
Press, Cambridge, 2004.
String theory:
-
Barton Zwiebach, A First Course in String Theory, Cambridge U. Press,
Cambridge, 2004. (The best easy introduction.)
-
Katrin Becker, Melanie Becker and John H. Schwartz, String
Theory and M-Theory: A Modern Introduction, Cambridge U. Press,
Cambridge, 2007. (A more detailed introduction.)
-
Michael B. Green, John H. Schwarz and Edward Witten,
Superstring Theory (2
volumes), Cambridge U. Press, Cambridge, 1987. (The old testament.)
-
Joseph Polchinski, String Theory (2 volumes),
Cambridge U. Press, Cambridge, 1998. (The new testament —
he's got branes.)
Math is a much more diverse subject than physics, in a way: there
are lots of branches you can learn without needing to know other
branches first... though you only deeply understand a
subject after you see how it relates to all the others!
After basic schooling, the customary track through math starts with
a bit of:
and
To do physics you need calculus, multivariable calculus
and linear algebra:
To do physics you also need to know about ordinary differential
equations, which rely on calculus:
Most of the fundamental laws of physics are partial differential
equations; these rely on multivariable calculus:
Multivariable calculus is a prerequisite for complex analysis, which
is like 'calculus using complex numbers':
Calculus is a prerequisite for real analysis, which you can think of
as 'heavy-duty calculus':
To do topology, which is the study of general spaces,
it's good to already know the concept of
'continuous function', which you may have seen in calculus.
You can study set theory and logic without any of the previously
listed subjects:
Linear algebra is a good warmup for abstract algebra, where you
leave the real and complex numbers behind and consider more general
number systems:
not necessarily in exactly this order. Proofs become very important
at this stage. You need to know a little set theory and logic to
really understand what a proof is, but you don't even need calculus to
get started on number theory:
From then on, the study of math branches out into a dizzying variety
of more advanced topics! It's hard to get the "big picture" of
mathematics until you've gone fairly far into it; indeed, the more I
learn, the more I laugh at my previous pathetically naive ideas of
what math is "all about". But if you want a glimpse, try
these books:
-
F. William Lawvere and Stephen H. Schanuel,
Conceptual Mathematics: a First Introduction to Categories,
Cambridge University Press, 1997. (A great place to start.)
-
Saunders Mac Lane, Mathematics, Form and Function, Springer,
New York, 1986. (More advanced.)
- Jean Dieudonne, A Panorama of Pure Mathematics, as seen by
N. Bourbaki, translated by I.G. Macdonald, Academic Press, 1982.
(Very advanced — best if you know a lot of math already.
Beware: many people disagree with Bourbaki's outlook.)
I haven't decided on my favorite books on all the basic math topics,
but here are a few. In this list I'm trying to pick the
clearest books I know, not the deepest ones — you'll
want to dig deeper later:
Finite mathematics (combinatorics):
-
Arthur T. Benjamin and Jennifer Quinn, Proofs that Really Count:
The Art of Combinatorial Proof, The Mathematical Association of
America, 2003.
-
Ronald L. Graham, Donald Knuth, and Oren Patashnik,
Concrete Mathematics, Addison-Wesley, Reading, Massachusetts,
1994. (Too advanced for a first course in finite mathematics, but
this book is fun — quirky, full of jokes, it'll teach you
tricks for counting stuff that will blow your friends minds!)
Probability theory:
Calculus:
Multivariable calculus:
Linear algebra:
This is a great linear algebra book if you want to understand
the subject thoroughly:
These books are probably easier, and they're free online:
Ordinary differential equations — some free online books:
Partial differential equations — some free online books:
Set theory and logic:
-
Herbert B. Enderton, Elements of Set Theory, Academic Press,
New York, 1977.
-
Herbert B. Enderton, A Mathematical Introduction to Logic,
Academic Press, New York, 2000.
-
F. William Lawvere and Robert Rosebrugh, Sets for Mathematics,
Cambridge U. Press, Cambridge, 2002. (An unorthodox choice, since this
book takes an approach based on category theory instead of the old-fashioned
Zermelo-Fraenkel axioms. But this is the wave of the future, so
you might as well hop on now!)
Complex analysis:
-
George Cain, Complex Analysis, available free online at
http://www.math.gatech.edu/~cain/winter99/complex.html.
(How can you not like free online?)
-
James Ward Brown and Ruel V. Churchill, Complex Variables
and Applications, McGraw-Hill, New York, 2003.
(A practical introduction to complex analysis.)
-
Serge Lang, Complex Analysis, Springer, Berlin, 1999.
(More advanced.)
Real analysis:
Topology:
-
James R. Munkres, Topology, James R. Munkres, Prentice Hall,
New York, 1999.
-
Lynn Arthur Steen and J. Arthur Seebach, Jr.,
Counterexamples in Topology, Dover, New York, 1995. (It's fun to see
how crazy topological spaces can get: also, counterexamples help you
understand definitions and theorems. But, don't get fooled into thinking
this stuff is the point of topology!)
Abstract algebra:
I didn't like abstract algebra as an undergrad. Now I love it!
Textbooks that seem pleasant now seemed dry as dust back then. So,
I'm not confident that I could recommend an all-around textbook on
algebra that my earlier self would have enjoyed. But, I would have
liked these:
-
Hermann Weyl, Symmetry, Princeton University Press, Princeton,
New Jersey, 1983. (Before diving into group theory, find out why it's
fun.)
-
Ian Stewart, Galois Theory, 3rd edition, Chapman and Hall,
New York, 2004. (A fun-filled introduction to a wonderful application
of group theory that's often explained very badly.)
-
Tom Leinster, Galois Theory, 2024. (Good to read after Stewart's book, and it comes
with videos and many problems and multiple choice questions.)
Number theory:
These are elementary textbooks; for more advanced ones
read on further.
-
George E. Andews, Number Theory, Dover, New York, 1994.
(A good elementary introduction; don't buy the Kindle version of
this edition since the equations are tiny.)
-
Joseph Silverman, Friendly Introduction to Number Theory,
Pearson, 2017. (Doesn't require any advanced mathematics, not even
calculus.)
-
Martin H. Weissman, An Illustrated Theory of Numbers,
American Mathematical Society, Providence, Rhode Island, 2017.
(This reveals the oft-hidden visual side of number theory.)
More Advanced Math
I'll start with some books on mathematical physics, because that's been
one of my favorite subjects for a long time.
Out of laziness, I'll assume you're already somewhat comfortable with
the topics listed above — yes, I know that requires about 4 years of
full-time work! —l and I'll pick up from there. Here's a good place to
start:
- Paul Bamberg and Shlomo Sternberg,
A Course of Mathematics for Students of Physics,
Cambridge University, Cambridge, 1982. (A good basic introduction
to modern math, actually.)
It's also good to get ahold of these books and keep referring
to them as needed:
-
Robert Geroch, Mathematical Physics, University of Chicago
Press, Chicago, 1985.
- Yvonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret
Dillard-Bleick, Analysis, Manifolds, and Physics (2 volumes),
North-Holland, 1982 and 1989.
Here's a free online reference book that's 787 pages long:
Here are my favorite books on various special topics:
Group theory in physics:
-
Shlomo Sternberg, Group Theory and Physics, Cambridge University Press,
1994.
- Robert Hermann, Lie Groups for Physicists, Benjamin-Cummings, 1966.
- George Mackey, Unitary Group Representations in Physics,
Probability, and Number Theory, Addison-Wesley, Redwood City,
California, 1989.
Lie groups, Lie algebras and their representations —
in rough order of increasing sophistication:
- Brian Hall, Lie Groups, Lie Algebras, and Representations,
Springer, Berlin, 2003.
-
William Fulton and Joe Harris, Representation Theory — a First
Course, Springer, Berlin, 1991.
(A friendly introduction to finite groups, Lie groups, Lie algebras and their
representations, including the classification of simple Lie algebras.
One great thing is that it has many pictures of root systems, and
works slowly up a ladder of examples of these before blasting the reader
with abstract generalities.)
- J. Frank Adams, Lectures on Lie Groups, University of Chicago
Press, Chicago, 2004.
(A very elegant introduction to the theory of semisimple Lie groups
and their representations, without the morass of notation that tends
to plague this subject. But it's a bit terse, so you may need to look
at other books to see what's really going on in here!)
-
Daniel Bump, Lie Groups, Springer, Berlin, 2004.
(A friendly tour of the vast and fascinating panorama of mathematics
surrounding groups, starting from really basic stuff and working on up
to advanced topics. The nice thing is that it explains stuff without
feeling the need to prove every statement, so it can cover more territory.)
Geometry and topology for physicists — in rough order of
increasing sophistication:
-
Gregory L. Naber, Topology, Geometry and Gauge Fields: Foundations,
Springer, Berlin, 1997.
-
Chris Isham, Modern Differential Geometry for Physicists,
World Scientific Press, Singapore, 1999. (Isham is an expert on
general relativity so this is especially good if you want to study that.)
-
Harley Flanders, Differential Forms with Applications to the Physical
Sciences, Dover, New York, 1989. (Everyone has to learn differential
forms eventually, and this is a pretty good place to do it.)
-
Charles Nash and Siddhartha Sen,
Topology and Geometry for Physicists, Academic
Press, 1983. (This emphasizes the physics motivations... it's not
quite as precise at points.)
- Mikio Nakahara, Geometry, Topology, and Physics, A. Hilger, New York,
1990. (More advanced.)
- Charles Nash, Differential Topology and Quantum Field Theory,
Academic Press, 1991. (Still more advanced — essential if you want
to understand what Witten is up to.)
Geometry and topology, straight up:
-
Victor Guillemin and Alan
Pollack, Differential Topology, Prentice-Hall,
Englewood Cliffs, 1974.
-
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,
Modern Geometry — Methods and Applications, 3 volumes,
Springer, Berlin, 1990. (Lots of examples, great for building
intuition, some mistakes here and there. The third volume is an
excellent course on algebraic topology from a geometrical viewpoint.)
Algebraic topology:
Geometrical aspects of classical mechanics:
- Vladimir I. Arnol'd, Mathematical Methods of Classical Mechanics,
translated by K. Vogtmann and A. Weinstein, 2nd edition, Springer,
Berlin, 1989. (The
appendices are somewhat more advanced and cover all sorts of nifty
topics.)
Analysis and its applications to quantum physics:
-
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics
(4 volumes), Academic Press, 1980.
And moving on to pure mathematics...
Knot theory:
- Louis Kauffman, On Knots, Princeton U. Press, Princeton, 1987.
- Louis Kauffman, Knots and Physics, World Scientific, Singapore, 1991.
- Dale Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976.
Homological algebra:
-
Joseph Rotman, An Introduction to Homological Algebra,
Academic Press, New York, 1979. (A good introduction to a branch of
math that's fundamental to algebraic topology. Try to get the 1979
edition, since some newer ones are much fatter, packed with
distracting stuff.)
-
Charles Weibel, An Introduction to Homological Algebra,
Cambridge U. Press, Cambridge, 1994. (Despite having the same title
as the previous book, this goes into many more advanced topics.)
Ring theory:
-
Frank W. Anderson and Kent R. Fuller, Rings and Categories of
Modules, Springer, 1992.
Combinatorics:
Algebraic geometry:
I found Hartshorne's famous book quite off-putting the first ten times
I tried to read it. I think it's better to start by getting to know
some 'classical' algebraic geometry so you see why the subject is
interesting and why it's called 'geometry' before moving on to
delightful modern abstractions like schemes. So, start with this
introduction:
-
Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen
and William Traves, An Invitation to Algebraic Geometry, Springer,
Berlin, 2004.
Then try these:
-
Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third
edition, Springer, 2013.
-
David Eisenbud and Joseph Harris, The Geometry of Schemes, Springer,
2006.
-
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry,
1994. (Especially nice if you like complex analysis, differential
geometry and de Rham theory.)
Number theory:
-
Kenneth Ireland and Keith Rosen, A Classical Introduction to
Modern Number Theory, second edition, Springer, 1998.
(A good way to catch up on some classic results in number theory
while getting a taste of modern methods.)
-
Yu. I. Manin and Alexei A. Panchishkin, Introduction to Modern
Number Theory: Fundamental Problems, Ideas and Theories, Springer,
2007. (Much more hard-hitting, but a very useful overview of what
modern number theory is like.)
-
Jürgen Neukirch, Algebraic Number Theory, Springer,
2010. (A friendly introduction to class field theory.)
Category theory:
-
Eugenia Cheng, The Joy of Abstraction: an Exploration of Math,
Category Theory, and Life, Cambridge University Press, 2022.
(A very enjoyable and easy way to get started on category theory,
leading up to the Yoneda Lemma.)
-
Brendan Fong and David Spivak, Seven Sketches in Compositionality:
An Invitation to Applied Category Theory, Cambridge U. Press,
2019. (A good introduction to category theory through applications;
available free on the arXiv.
Also see their videos and my lectures.)
-
Tom Leinster, Basic Category Theory, Cambridge Studies in
Advanced Mathematics, Vol. 143, Cambridge U. Press, 2014. Also
available for free on the
arXiv. (An introduction for beginners that focuses on three key
concepts and how they're related: adjoint functor, representable
functors, and limits.)
- Emily Riehl, Category Theory in
Context, Dover, New York, 2016. Also available for free on her
website. (More advanced. As the title suggests, this gives many
examples of how category theory is applied to other subjects in math.)
I have always imagined that Paradise will be a kind of library. -
Jorge Luis Borges
baez@math.removethis.ucr.andthis.edu
© 2019 John Baez
