Hi all, I'm a bit new here. I was unsure where to post this but this seems most appropriate to me.

Getting down to it, I'm interested in a number of topics in advanced physics. Specifically, I'd like to begin to do a bit of self-guided study on special/general relativity and eventually QM and unified field theory. My background in physics isn't enormous, but I have taken college courses in classical mechanics and electromagnetism/optics (physics I and II, calculus treatment). I've also had 3 semesters of calculus.

I can already hear someone practically screaming "MATH MATH MATH!" What math topics should I start with? I'm assuming differential equations is the next step, but I'm not sure where to go from there. Vector analysis? What are my next steps as far as math and also physics? If someone could offer me a sort of curriculum guideline it would really help me out.

Thanks for making this a great place to ask questions!

Quantum Mechanics "is" linear algebra, so you'll want to know all the general rules of Matrices, Vector spaces, Hermitian matrices, conjugate transpose, etc.
You'll need partial differential equations for E&M and a full understanding of vector calculus and complex variables.

While it seems universities focus on methods of Analysis (calculus), I think that knowledge of numerical methods is probably at least as important when attempting to solve "real" physical problems.

If it helps, I'm currently self-studying upper level physics as well. I'm an applied mathematics major with physics minor (I wanted to be a physics major but each semester had conflicts with physics courses and I ended up getting further ahead in math...) and my math background is as follows:
Biostatistics
Calculus I, II, and III
Linear Algebra
Differential Equations and matrix algebra
Mathematical proofs
Vector Calculus and complex variables (fall)
Numerical Analysis (fall)

Firstly, in terms of difficulty, you'll probably want to learn QM & special relativity first AND THEN GR and unified field theory. The math basis required for QM and SR would be something like:

calculus 1 and 2
vector calculus
differential equations
linear algebra
mathematical physics 1 and 2

If you wanna do GR you'll also need something like:
differential geometry and tensor analysis

If you wanna do string theory and such well... I'd say worry about that after you've got the others under your belt (btw, if you wanna know physics you will also need to know electromagnetics, thermodynamics, statistical mechanics and classical mechanics and you will need all of them).

In terms of textbooks (in chronological order) I'd say something like:

-Stewart's Calculus
-An introductory linear algebra book
-A mathematical physics book like Mary Boas'

then you can get into the physics. The classic intro texts are (although I'm sure every person will have a different favorite but these are among the most commonly used):
-Classical mechanics -> Goldstein
-Introductory Quantum mechanics -> Griffiths
-Electromagnetics -> Griffiths (same guy, different book, he's actually written like 5 books) also this will be where you learn basic SR
-Something like Thermal Physics by Kittel or Schroeder (different books but same title)

Once you've got that i'd say come back and ask for the next step but I have a feeling that's more than enough to keep you busy for quite awhile

diazona
Homework Helper
-Classical mechanics -> Goldstein
-Introductory Quantum mechanics -> Griffiths
-Electromagnetics -> Griffiths (same guy, different book, he's actually written like 5 books) also this will be where you learn basic SR
3 books actually: QM, EM and particle physics ;-) Seriously though, I would recommend going through SR at least once before you get into E&M, because it's a tricky thing to get used to. I learned it as part of introductory mechanics.
-Something like Thermal Physics by Kittel or Schroeder (different books but same title)
I highly recommend Schroeder, if you want to actually understand what you're reading. I'd go so far as to call it the Griffiths of statistical mechanics.

A bit of an overview here mrb; Be careful not to follow the path of entirely math led physics. Math is just a tool.

My advice is try for the conceptual undersatanding first. You'll find that SR, and current physics in general is full of paradox. Read things like 'The Road to Reality', and 'The Trouble with Physics'. You'll find the more you find out the more you relaise how little we know. Don't beleive the guys who are happy in their own comfort zone, even Einstein on his deathbed confirmed there was something wrong in relativity. Look at the history leading up to it, and what he actually said (i.e. the Leyden talk) and you may work out what.

Then you may stand a chance of making a contribution to the future of physics and mankind.
But if you just prefer doing sums don't let me stop you.

ZapperZ
Staff Emeritus
A bit of an overview here mrb; Be careful not to follow the path of entirely math led physics. Math is just a tool.

My advice is try for the conceptual undersatanding first. You'll find that SR, and current physics in general is full of paradox. Read things like 'The Road to Reality', and 'The Trouble with Physics'. You'll find the more you find out the more you relaise how little we know. Don't beleive the guys who are happy in their own comfort zone, even Einstein on his deathbed confirmed there was something wrong in relativity. Look at the history leading up to it, and what he actually said (i.e. the Leyden talk) and you may work out what.

Then you may stand a chance of making a contribution to the future of physics and mankind.
But if you just prefer doing sums don't let me stop you.
But your suggestion is nothing more than asking someone to have a superficial understanding of physics. These are books ABOUT physics, not books on physics. There's a difference! No one reading those books alone will be able to solve physics problems and have a "chance of making a contribution to the future of physics and mankind". That's a gross overstatement! Did this year's Nobel Prize winners in physics read those books that somehow triggered their enormous contribution to physics?

The OP is asking for actual physics resources to understand advanced topics in physics. In the realm of mathematical preparations, I've recommended the Mary Boas's text "Mathematical Methods in the Physical Sciences" several times on here (do a search). If one needs a quick survey of many of the relevant "tools" of mathematics that a typical physicist needs, then one can do no wrong in getting this book.

Zz.

Staff Emeritus
2019 Award
even Einstein on his deathbed confirmed there was something wrong in relativity.

Einstein on his deathbed confirmed there was something wrong in relativity.
Einstein spoke more on his "deathbed" than most men speak in a lifetime.

If he really made as many "deathbed statements" as I've seen claimed, he would have needed to spend a good 35 years dying in bed and giving lectures.

Books like "Road to Reality" and "Fabric of the Cosmos" are little more than public relation vehicles for people working on unified theories who want to garner support and funding (and make some quick cash) by circumventing any sort of academic or informed discourse and just rabble rousing the public who has no clue what they're talking about.

Far more importantly, people who read those books often get the VERY WRONG, idea that most physicists do this kind of work. Maybe 1% of physicists work on grand unified theories the VAST majority never deal with it, it's kind of a niche field. By the numbers, most physicists work in condensed matter physics.

In general, when you start your undergrad everyone wants to do string theory or some such. What most people find as they continue their undergrad is that it's either really not as interesting as they thought it was, or it's just not for them and by the end of undergrad most physics students are interested in something else entirely (condensed matter, computational, materials, polymers, astro, etc.)

And by the way, there's no "proof by alleged quote from really famous guy". Who or what said what about what is not particularly relevant to any form of logical and rigorous inquiry.

OK sorry, rather overstated deathbed bit for brevity. Don't want to hijac mrb's thread, and never rely on quotes as 'proof' of anything, but;

We all must know AE's comment that, effectively, if there was a quantum field SR would not work (look it up!), and By '55, after Feynman et al, with overwhelming evidence of QM and QED that wouldn't fit SR, he confirmed acceptance QM MIGHT be right by spending most of his time on unification (even on the last day in bed at Princeton). If you read German you'll have read this from late '54; (I've just chased it down in english at APSnews/200512/history.cfm)

"I must seem like an ostrich who forever buries its (sic) head in the relativistic sand in order not to face the evil quanta."

OK, it only confirms he knew there was something wrong somewhere.
(I won't repeat his quote about math!). mrb - I'm just saying I feel you ought to get a proper overview of anything before you get fully embedded in the fine detail. It's all about the risk of not seeing the trees for the wood.

1954 predates quantum field theory (which unites quantum mechanics and SR) by quite a bit but I've never heard anything about QED and SR having conflicts

I am pretty sure quantum field theory is an extension of quantum physics with relativity. In basic terms, the Dirac equation is just a relativistic formulation of Schrodinger's equation. And I would say that so far, undergrad physics has exceeded my wildest imaginations, especially with regards to electromagnetism, quantum physics and solid state physics.

Staff Emeritus
2019 Award
But your suggestion is nothing more than asking someone to have a superficial understanding of physics.
Exactly.

You'll find that SR, and current physics in general is full of paradox.
I note that you haven't actually explained what this paradox is - or where it is. Reread what ZapperZ wrote about superficial understandings.

I've just chased it down in english at APSnews/200512/history.cfm)

"I must seem like an ostrich who forever buries its (sic) head in the relativistic sand in order not to face the evil quanta."
Did you read the whole article? It discusses Einstein's desire to come up with a more expansive theory that would have QM as one of its consequences. Today we know he was unsuccessful at it, and what we have learned since then makes it likely Einstein was barking up the wrong tree, but it most certainly does not say Einstein was saying - as you claim he was - that something is wrong in relativity.

PS "its" without the apostrophe is possessive. With the apostrophe, it's a contraction for "it is."

ZapperZ
Staff Emeritus
Before we go overboard with this, take note that the OP hasn't responded, or come back yet after that one post. We may be busying correcting each other and the OP would not get any of these.

Zz.

Before we go overboard with this, take note that the OP hasn't responded, or come back yet after that one post. We may be busying correcting each other and the OP would not get any of these.

Zz.
Haha, I'm back. Thanks for all the replies folks. I think I'll check out that book, The Road to Reality. But eventually I'd like to get into the math as well. I have Boas book on mathematical methods, so I'll get into that after the road to reality. Has anyone else read taht book? What did you think of it?

Thanks guys.

Well done mrb. But you (and Zz) should note I only said not 'entirely' maths, ..and overview 'first'.

Maveric has proved my point by appearing not to know the relevant part of QM first came along only just after SR! And by not knowing about the conflicts between QED and SR.

and
'I am pretty sure quantum field theory is an extension of quantum physics with relativity. In basic terms, the Dirac equation is just a relativistic formulation of Schrodinger's equation. And I would say that so far, undergrad physics has exceeded my wildest imaginations, especially with regards to electromagnetism, quantum physics and solid state physics.'

be aware; QFT is NOT QP with relativity, indeed au contraire!! - and much more homework on Diracs 'relativistic electron' is required.

Dont be fooled when people tell you all the paradoxes in physics are resolved, but be honest when finding your own comfort zone. Hope you enjoy it as much as Maveric, and don't be afraid of looking at AE's quotes, they don't prove anything but they can give you insight. Best of luck.

Ivan Seeking
Staff Emeritus
Gold Member
In the realm of mathematical preparations, I've recommended the Mary Boas's text "Mathematical Methods in the Physical Sciences" several times on here (do a search). If one needs a quick survey of many of the relevant "tools" of mathematics that a typical physicist needs, then one can do no wrong in getting this book.

Zz.
My copy is never more than ten feet away from my desk.

I didn't realize that anyone else [generally speaking] uses this book.

My copy is never more than ten feet away from my desk.

I didn't realize that anyone else [generally speaking] uses this book.
Mine is typically right next to whatever Physics or Math book I'm studying...it was given to me by a friend and I can't begin to count the number of times it has helped clear up a confusion or drive home a point about a subject. I have a few books on the mathematics of the physical sciences, but none of them cover the range of topics with as much completeness (while maintaining brevity) of Boas'.

Ivan Seeking
Staff Emeritus