I would argue that it is actually the mathematicians who make the better theoretical physicists. A physics degree rarely prepares you for the modern mathematical constructions used in theoretical models.
Of course, you'll need to study physics along with your fibre bundles. :-)
What courses do you think would be needed for the physics people?
I'll be honest, the mathematics I've studied has been self taught for the explicit purpose of understanding curvature and covariant derivatives for GR. What mathematics would be required for loop quantum gravity, I really don't know.
But, at a minimum, I would say that courses in topology, group theory, and differential geometry are a requirement for a foundation.
this are kind of math classes i'd do - to add to the physics
anything with a [x] would be normally taken for a physics degree
[x] Math 151 Calculus I
[x] Math 152 Calculus II
[x] Math 251 Calculus III
[x] Math 252 Vector Calculus I
Math 313 Vector Calculus II / Differential Geometry
Math 466 Tensor Analysis [needs Differential Geometry]
Math 471 Special Relativity [needs Differential Geometry and Butkov] [Butkov needs Diff Eqs and Griffith EM]
Analysis and Topology
Math 242 Intro to Analysis
Math 320 Theory of Convergence [aka Advanced Calculus of One Variable]
Math 425 Introduction to Metric Spaces
Math 426 Introduction to Lebesque Theory
Math 444 Topology
[x] Math 310 Introduction to Ordinary Differential Equations
[x - maybe honours] Math 314 Boundary Value Problems
Math 415 Ordinary Differential Equations [needs Complex Analysis]
Math 418 Partial Differential Equations [needs Differential Geometry]
Math 419 Linear Analysis [needs Theory of Convergence]
Math 467 Vibrations [needs Symon]
Math 470 Variational Calculus [needs Symon and Differential Geometry]
[x - maybe honors] Math 322 Complex Analysis
Math 424 Applications of Complex Analysis
[x] Math 232 Elementary Linear Algebra
Math 438 Linear Algebra
Math 439 Introduction to Algebraic Systems [aka Abstract Algebra]
Fluid Mechanics [fluid motion/air motion/turbulence]
Math 362 Fluid Mechanics I [needs Vector Calculus and Symon]
Math 462 Fluid Mechanics II [needs Boundary Value Problems]
Continuum Mechanics [aka deformation/stress/elasticity]
Math 361 Mechanics of Deformable Media [needs Vector Calculus and Engineering Dynamics]
Math 468 Continuum Mechanics [needs Differential Geometry and Boundary Value Problems]
Probability and Statistics
Math 272 Introduction to Probability and Statistics
Math 387 Introduction to Stochastic Processes
Math 316 Numerical Analysis I [needs Fortran or PL/I]
Math 416 Numerical Analysis II [needs Differential Equations]
That's a lot said. I wouldn't mention course numbers because they're different for every university.
There really is no prerequisite for topology, you could read that from a good book like Mendelsohn right now. Linear Algebra is certainly a requirement. and Diffy G will require a calculus sequence. PDEs do not require differential geometry. It's the other way 'round.
I forgot to mention measure theory. that would include Lebesgue integration along with path integration as done in QFT.
I only included course numbers since knowing something is a 400 class is usually best for 4th year courses, and 100 level classes are for first year folks..
oh and i like the look of those numbers too lol
Well most topology texts would demand a good solid 6 months of Analysis. Some are readable without any prerequisites other than maturity, and you learn your analysis as you go along.
As for some PDE courses, they 'recommend' differential geometry...
Real Analysis by Royden would have Measure Theory, and well most people would be doing that stuff for probability, and maybe Fourier stuff.... and it's more like people are just being pushed to take 'advanced Real Analysis' so they can take Functional Analysis in Grad School....
you get pummelled with some Mathematical Physics thing like the 4 volume Princeton Set Reed and Simon, and with Reed you can push into where i think this 'math' is leading some
'path integrals in Quantum Mechanics'
as one person said it:
[One of the strangest and at the time most difficult topics I ever tried to understand. Path Integrals. Actually a part of what's called "Functional Analysis". But now after some "miraculous conversion" which has taken place in my psyche these path integrals seem not so bad after all.]
and this is where it might lead into:
Quantum Physics: A Functional Integral Point of View - Glimm and Jaffe
[Constructive Quantum Field Theory by its founders]
[Not well written]
[This book is unnecessarily hard to read]
[You need Book 1 and 2 of Reed's Functional Analysis first]
[People who complain that this book is uneccessarily hard to read probably have bought it for the wrong reasons, guided by its deceptive title. This is NOT a standard book on quantum mechanics via path integrals, as is the marvellous book by Feynman and Hibbs, among others. This is the bible on Constructive QFT (CQFT),
the most recent of all attempts to put QFT on a sound mathematical basis, written by two of its founders. This IS hard physics AND hard math - Wiener integrals, unbounded operators, and so on.]
[Although the book is reasonably self-contained, it is strongly recommended to have a fair grounding on analysis, quantum mechanics and basic quantum field theory. The last part of the book (chapters 13-23) is meant for pros, requiring more maturity.]
[Anyway, it's tough to find similar texts on this subject. The reader who is interested in this field of research and have some guts for tough mathematical physics certainly will be rewarded. This tome surely stands on its subject in the same footing that Haag's "Local Quantum Physics" on Algebraic QFT]
another book that shows a similar mathematical path
when you ask, what the heck is FUnctional Analysis used for
or i read that von Neumann studied the math before going into Quantum stuff
Quantum Mechanics in Hilbert Space - Second Edition - Eduard Prugovecki - Academic Press 1981/Dover
[A rigorous, critical presentation of the basic mathematics of nonrelativistic quantum mechanics, this text is suitable for courses in functional analysis at the advanced undergraduate and graduate levels. Its readable, self
contained form is accessible to students without an extensive mathematical background. Numerous exercises include hints and solutions. Reprint of the Academic Press, New York, 1981 edition.]
[This book is for the careful and diligent student that wants to get a firm basis on the mathematical foundations of the subject without wanting to become a fully blown mathematical physicist. Take it or not, it is up to you, but it is the modern alternative to the classic book by von Neumann]
another book to hint at the higher math being useful
Mathematical Physics - Robert Geroch - University of Chicago 1985
[This is one of my favorite books. I think this book should be mandatory reading for physics students; it'll teach them a lot about the proper use of mathematics, and the benefits of using math correctly.]
[I first read this excellent book about six years ago when I was beginning to realize that my undergraduate physics education wasn't going to teach me the math I wanted to know, especially since my main interest in physics was of a more theoretical bent. This book provides an excellent outline of many of the mathematical concepts that are rarely covered (or at least rarely covered well) in most physics curriculums. Running the gamut from category theory to basic group theory to algebraic topology and beyond, this book provides much of the basic mathematical framework that many modern physical theories rely on. Highly recommended.]
[Geroch's book contains a broad survey of abstract algebra, topology, and functional analysis, and it does a wonderful job at motivating mathematical definitions and constructions. Surprisingly, since Geroch is an expert, it contains no differential geometry. Also, its layout is abominable. - George Jones]
Methods of Modern Mathematical Physics - Four Volume Set - 400 pages - Academic Press 1981 - 400 pages - for book one - Michael Reed and Barry Simon
a. Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis - Reed
b. Fourier Analysis, Self-Adjointness - Vol. 2 - Reed
c. Scattering Theory (Methods of Modern Mathematical Physics, Vol. 3) - Simon
d. Methods of Modern Mathematical Physics: IV Analysis of Operators - Simon
[Mathematical methods in physics: Emphasis on foundations: quantum mechanics, Hilbert space, symmetry, unitary representations.]
[The best functional analysis book for beginners, in my opinion. It is written for people that are interested in functional analysis as a tool for differential equations. What makes it different from other books on this subject are the numerous examples and applications to differential equations. Highly recommended.]
[could go with Courant's Methods of Mathematical Physics and Conway's book on Functional Analysis]
simply said, just read books on mathematical physics
if you take undergrad quantum mechanics
maybe in grad school, extra linear algebra and analysis helps...
and people can do higher up 'path integrals' and 'QFT aka Quantum Field Theory'
yeah, its possible. Go research, nothings stopping you. You dont even need an education, just an imagination, and a firm grip on reality at the same time ha.
Yes of course , but You simply can't do truely interesting stuff unless you know Quantum field theory and the gravitational force if you want to do string theory . You have to be comfortable with hand-waving not rigorously defining and proving every thing .
you're assuming everyone wants to do string theory.
[we're already off to a BRUTAL start if you need QFT]
[solid background in QM and Special Relativity - yow]
[maybe adding - Tensors - Group Theory - Differential Geometry - Lie Theory]
a. one Quantum Text [Eisberg - Liboff - Griffith]
b. one Grad Level QM Text [like Sakurai]
group theory like books
b. Group Theory in Quantum Mechanics - Volker Heine - Dover 1993
Baby Books on QFT
a. A First Book of QFT - Lahiri - CRC Press
b. A Modern Introduction to Quantum Field Theory - Michele Maggiore - Oxford
c. Quantum Field Theory for the Gifted Amateur - Tom Lancaster - Oxford 2014
d. From Special Relativity to Feynman Diagrams: A Course of Theoretical Particle Physics for Beginners - D'Auria & Trigiante
[gives all the prerequisites for QFT - Lie groups, relativistic electrodynamics, Lagrange & Hamiltonian mechanics]
e. Problem book in QFT - Radovanovic
f. Quantum Field Theory and the Standard Model - Matthew D. Schwartz - Cambridge 2013
g. Student Friendly Quantum Field Theory - Robert D. Klauber
h. Mandl and Shaw - Quantum Field Theory
i. Gauge Theories in Particle Physics - Two Volume - IJR Aitchison
[Volume I: From Relativistic Quantum Mechanics to QED]
[Volume II: A Practical Introduction : Non-Abelian Gauge Theories : Qcd and the Electoweak Theory]
[Once you've mastered QM, you might want to dip a toe into the Quantum Field Theory waters. This is a gentle introduction.]
j. Introduction to Quantum Field Theory - V.G. Kiselev - CRC Press 2000
k. Gauge Theory and Variational Principles - David Bleecker - Dover
[This text serves as an introduction to the QFT and guage theories recast in the 'modern' mathematical setting of differential geometry. This book is only 167 pages long. Although self-contained I highly recommend the reader have a working knowledge of QFT and at least an introductory course in GR. The mathematical tools of the reader should include a course in analysis on manifolds at the Spivak level or higher, acquintance with fibre bundles and basic lie groups.]
[i said it was gonna be BRUTAL]
old style people got QFT from Bjorken and Drell
new style poeple got QFT from Peskin
a. Itzykson and Zuber - Quantum Field Theory - 1980
b. Peskin & Schroder - the standard QFT textbook
c. Ryder - Quantum Field Theory - Cambridge 1984
d. An Interpretive Introduction to Quantum Field Theory - Paul Teller - Princeton 1995
d. Local Quantum Physics: Fields, Particles, Algebras - R. Haag
e. Anomalies in Quantum Field Theory - Reinhold A. Bertlmann - Oxford 2001
g. Quantum Field Theory in a Nutshell - A. Zee - Princeton 2000s?
h.Tom Banks - [very concise and inaccessible to beginners]
i. Mark Srednicki - Quantum Field Theory - Cambridge 2007
[a higher level of abstraction and is great for a second book]
[This accessible and conceptually structured introduction to quantum field theory will be of value not only to beginning students but also to practicing physicists interested in learning or reviewing specific topics. The book is organized in a modular fashion, which makes it easy to extract the basic information relevant to the reader's area(s) of interest. The material is presented in an intuitively clear and informal style.]
j. Weinberg's 3 volumes - Cambridge 1995
[notoriously difficult to learn from, but still the reference for certain topics]
[Read Ryder/Peskin/Zee first]
k. Supersymmetric Gauge Field Theory and String Theory - Bailin and Love
l. Wess and Bagger - Supersymmetry and Supergravity - Princeton
m. Field Theory: A Modern Primer - Pierre Raymond - Westview Press
n. Relativistic Quantum Mechanics/Relativistic Quantum Fields - Two Volumes - Bjorken and Drell - McGraw-Hill 1964/1965
Book 1 - Relativistic Quantum Mechanics - McGraw-Hill 1964
Book 2 - Relativistic Quantum Fields - McGraw-Hill 1965
[Still used at Berkerley 15 years ago for QFT]
For Stupid Strings
a. Superstring Theory - volume I and II - M.B. Green, J.H. Schwarz, E. Witten - Cambridge 1988
[Although these two volumes do not touch the important new developments in string theories they are still the best texts for the basics]
[GSW is slightly old now; it was written in 88 and doesn't contains a lot of the recent developments like duality and M-theory. It's still quite interesting to read; you can learn a lot about field theory by studying this book. And if you want to do string theory, you'll probably have to read it anyways.]
[considered the readable text on superstrings where polchinski isnt readable]
b. tring Theory and M-Theory: A Modern Introduction - Katrin Becker - Cambridge 2006
c. Michio Kaku - Strings, Conformal Fields and Topology - Springer 2000
Berkeley - probably did
a. Peskin & Schroeder
b.Wess and Bagger
d. Bialin and Love
f. Bjorken & Drell
THEN then go nuts with silly string, though i'd waste it more with
ADM Formalism or Kaluza-Klein
since Strings are nutty like the Stock Market crazies who do - Technical Analysis - and lots of hocus pocus
oh yeah none of these books are recommendations, just my taste for books
almost makes ya wanna swear off physics, i tell ya
and i thought oh man, do i really wanna add to the clutter of textbooks way too hard for 97% of humans who took 3 physics courses or less
but i thought i would say
a. QFT, oh man talk about a scary cliff for some people to jump off into, very tackle it as an undergrad
b. Superstrings with a background in QFT, usually done if you wish to take the slow route, but still brutal for what might be a theoretical waste of time. What happens if ADM Formalism opens up in 30 years because String Theory was just abstract weirdness that didnt do much?
QFT, oh man talk about a scary cliff for some people to jump off into, very few tackle it as an undergrad....
But at least from the above rant, you can say, oh i heard of Steve Weinberg with quarks and things like the electroweak theory and the unification of the forces and the Big Bang Theory stuff, when he was all famous in the early 1970s, and on NOVA endlessly....
WOAH. Easy, young feller. There's no need to quote every textbook under the Sun. I always pick just one that sounds good and build as I need to.
The question was about math for theoretical physics and QFT qualifies as that. If you're restricting yourself to only undergraduate mathematics, then the list goes: topology, group theory, linear algebra, calculus, ODEs, and PDEs. DiffyG is usually a grad course. But measure theory is a requirement. I learned just enough analysis to understand measure theory. But I suppose Analysis is an undergrad course.
Ryder's ok. Not on my list of faves. And start with Zwiebach for an overview of strings. If you still think they're a good idea (I'm not a fan), then I'm sure there are good recommendations.
To be sure, theoretical physics is not a walk in the park, but it certainly isn't "brutal." If it is, you're in the wrong field.
Well, i thought i would toss out all the books i knew that were recommended by people as stuff they *wished* they read before taking the usual textbook on QFT.
and well, some people do Particle Physics and then run into Quantum Field Theory
and *some* take QFT and then use that to play around in Stringy stuff.
depending on the text and course, Diff Geometry could just be squished on after Vector Calculus, and others want a whole year of Analysis [usually Rudin] and the teachers go ahem, 4th year maturity in this class buster....
half the classes teach Differential Geometry just like Calculus Four, others teach it like something one step down from Grad School....
two of the major books used
a. Spivak's book [actually it's a set of 5 books, but most people just buy 1 or 2 of them]
b. DoCarmo's book
Ryder's a lot better than what was before and still a top reference, and now there's always a book to read before the usual QFT books, so there's less confusion.
And well most courses are brutal, when you struggle with the prerequisites, or only skim the subject where it's more a 'chore' than a 'joy'.... For a lot of people, i suspect they get the best and worth of both!
Theoretical Physics? What is that?<br />
I'm sorry. I'll probably know someday, but for now, this goes beyond my freshman-in-highschool level education.
You could think of it this way....
Theoretical Physics is everything in physics if you take away the experiments.
The guy with the atom smasher is playing with toys...
the theory guy like Einstein, is playing with chalk and a blackboard.
what have you taken with math, and what have you taken with physics, and where would you like to go with 'more courses'?
some people get a Bachelors in one and then hop to the other for grad school and fill in the gaps for a few semesters..
and others take an extra year or two, and aim for a mathematical physics degree where it's pretty easy to shift to one or the other, depending where you wanna specialize.
Basically some people dont wanna do real analysis, and other people arent too keen on a lot of intermediate electromagnetism.
I like to say that if you get a physics degree and you take analysis and you take differential geometry, fairly early on, you can wander anywhere in both subjects.
When you have
a. vector calculus, and
b.a whole textbook on differential equations
most everything hard is pushed out of your way, and
c. with differential geometry,
you can tackle most any course.....