How Much Math Should a Theoretical Physicist Know?

[loonychune]

Hello.

What i find as a (first year undergraduate) theoretical physicist is this - THERE IS JUST NOT ENOUGH MATHS BEING TAUGHT... i am of the opinion that my maths ought to be as strong as possible, but in studying A. Beardon's 'Algebra & Geometry' I'm worried i might be heading slightly off course, for how necessary is it that I'm as rigorous as a mathematician?

I suppose what i ask is this: what ought i incorporate into my own degree from a maths degree??

I'm gettin along well with Riley, Hobson, Bence and also with Arfken, but don't really want to be too caught up in these books without doing maths-maths as well...

Any feedback would be much appreciated... merci

 

[math_owen]

What's going on loonychune or is it merci?

How much math is a question of how "Theoretical" this "man" wants to be. It is true that math and physics are different subjects. And as outlined by Feynman, physicists should use math only as a tool. But, they feed each other. And was Newton a mathematician who loved physics or was he a physicist who was exceptional apt at the mathematics of his time? Chicken or the egg?

There are 2 camps to mathematics proper, Bourbaki and Arnold. I'll let the applieds be for the moment. Bourbaki is dominant, and it has good cause to be. But, I suspect, that it's going to need to go through a revolution with the Arnold camp to create a new camp.

Often what makes the non-Bourbaki puke is their method. Personally I hate it. I'm not a machine. I'm all about rigor, but I don't see why we have to cut out the human side of the mathematics. I don't see why we can't give the motivation behind certain polishings of certain theories. It'll change. I'll help it change

Remember this. Even physicists are aware of this on some level. All these boundaries are human constructs and are entirely artificial outside of our minds. Physics is our best approximation to the truth of our reality. And mathematics is our best attempt at the most accurate reasoning of space, time, change, etc. that we can think of.

How much math? How much do you need? I suspect if you ever wish to have a conversation with Witten you'll know what mathematicians know. B/c he knows more math than many mathematicians

If your an experimentalist heading to IBM to help design future quantum toys, then most likely an undergraduate equivalent in mathematics will be over kill.

If you're like me, and heading towards mathematical physics. Well, like someone else in another post said, these types have to know an enormous amount of mathematics (as well as physics). Open Penrose's tomb, it'll give you a hint. And by the time you get there, that'll be dated, and the knowledge base will be doubled.

But, I'm told by the String Theorists at my school, that too much math for a theoretical physicist can be bad too. Why? B/c you might not be able to shut it down when you need too. Physicists take leaps, often creating bizarre ideas. Later mathematical physicists come behind them and turn it into something more rigorous.

Can you do both? Can you take leaps on the frontier? Can you think nitty gritty logical about every minute detail when you need to?

Odds are you'll know a lot of math. Even if you can't give me a formal definition of a manifold and tell me formally all it's properties, at the very least you'll be apt at manipulating one.

I checked out the book you listed on amazon. It looks fine. It gives a good overview of an undergraduate mathematics career without diving into great detail. You don't need a great book, a good one will do. What you need, is a great inner drive.

And oh, a plug for mathematics. It can be fun. If you can figure out that mathematics is not formal proofs. Its far different than that. That's just our attempt at cataloging (I agree it's a poor attempt).

Best of luck.

 

[loonychune]

Hey Owen... that's a smashin' reply - thanks a lot!
I have always said that i'd like my maths to be at least half as good as an undergraduate mathematicians compared to he and I at the same level - this has been revised to pretty much 'as good as' in the general scheme of things...

I'd appreciate a few specifics if anyone would like to add to this.. i.e. what such textbooks (solely mathematics texts) would suit me given where i am at (end of first year undergrad) and in the forthcoming years?

thankyou

 

[math_owen]

loonychune,

My wife humorously points out that merci is French for thank you, lol. I don't often leave the math and physics to explore other things. I should though.

Since you're new you probably haven't any idea what you want to study yet. So probably best is a broad background in mathematics for you.

If you end up in relativity and/or cosmology you'll be good at playing with manifolds and doing tensor calculus. For that you'll need to be good at almost all of the undergraduate mathematics; analysis, multi-dimensional analysis, topology.

For that you should supplement the courses texts with these if they don't already use them; Rudin's Principles of Mathematical Analysis, Analysis on Manifolds by Munkres, and Topology by Munkres. I would even judge your school on how close to those standards your classes were. No single dimensional measure could ever be truly accurate, but this is a good indicator.

If you go quantum algebra and stuff then I'm not completely sure because I have not taken quantum yet; but I think you're going to need to do advanced linear algebra and abstract algebra. I would suggest the books Linear Algebra by Hoffman and Kunze and Abstract Algebra by Dummit and Foote. I used Abstract Algebra by Gallian which is also good. As a physicist you should stay away from Lang and the Bourbaki books.

To help your calculus along there are classical texts. Namely Calculus vol 1 and 2 by Apostol and Calculus by Spivak. I own Spivak's and I think that's a top quality text. Though his Calculus on Manifolds is only good for problems in my opinion. He really went all out Bourbaki-style with that one. Instead use the Analysis on Manifolds by Munkre above.

You say no physics books, but I wish to justify my choices above. Currently I'm reading Wald's General Relativity. This book I'm learning GR from this summer to help my honors thesis in the Fall. The mathematical physicist who is mentoring me said this is the mathematician's preferred GR text. I can see why.

Being a math major I have a slight advantage. This book would be considered lax by a Bourbaki Mathematician, but to a physicist I'm guessing this is considered dense. I don't know of any physics only undergraduates who could read this, and from my friends in the masters program in physics I don't know if they would get much out of it if they hadn't read something like Gravity by Hartle first.

Now if you're thinking String Theory... My school happens to be excellent at it, and the running opinion is that String Theorists are Mathematicians. These guys honestly know at least a masters in math worth. They may not be able to pass our quals, but they're more than familiar with all the mathematics for a masters student.

Probably the best thing to do is take as much math as you can. That is take math until you can't stand it anymore or you can't handle it anymore. That would be my advice.

I'm told by my undergraduate physics friends that a course in probability and then a course in statistics is also very desirable. Especially if they're taught in an applied mathematics department.

The other common opinion I have heard from the undergraduate physics students is this, "don't learn mathematics from a physicist". This is probably true. But that is up for you to decide.

Best of luck.

 

[math_owen]

I have one other quick thing to add. In one of my classes we used a very impressive book. My school stopped using it though, because the students were complaining about it being too logic based.

Here's the quick reasoning. I did my first analysis class as an independent study (along with algebra too) at a community college. To frank and honest, the masters degrees who taught there were not the best at proof. So to say the very least I struggled fiercely with proof.

At my new school I took a "proof class". It was meant to be prior to analysis and algebra, but as usual I did it backwards. The benefits were huge though.

The book explained everything that use to kill me about how proofs move and the logic behind them. In my next analysis class there was an engineering ph.d student - a really smart one - and he was struggling hard with a long string of quantifiers. He concluded that we mathematicians pull stuff out of thin air and that analysis wasn't as "rigorous" as we thought it was b/c of the pulling out of thin air idea.

In short, it was a logic issue. All logic. He didn't understand the way quantifiers work when strung together with embedded if-then statements. That can be a pain if you don't understand them. I know.

So, especially for the non-mathematician, I recommended this book. And even for the mathematicians.

Proof, Logic and Conjecture, The Mathematician's Toolbox by Robert S. Wolf.

But, only after you've struggled with proof. It's not going to make much sense as my school learned if you haven't already struggled with and tried proof. But afterwards this will be the lighthouse in the storm you were looking for.

When you're struggling with proof I recommend, How to Read and Do Proofs by Daniel Solow.

This two books have been a cornerstone in my education so far, and I know I'll re-read them many more times.

It's the little things, the foundational things, that will kill you.

This turned out longer than I wanted it to be.

Best of Luck!

 

[WolfOfTheSteps]

I just wanted to say thanks for the post, math_owen. I'm an EE minoring in pure math, and while I've read a couple proof books, I'm always looking for more good ones. I just ordered a copy of your Solow recommendation. :)

 

[math_owen]

No problem at all. You're definitely going to like it. I feel akin with EE types. I was an electronic technician in the Navy and had received 2 years of electronics training. I know EE uses a ton of math. My electronics professor in college worked on some really mathematical intense stuff.

Here's a book he wrote that's at amazon. Impressive the level of mathematics he used.
https://www.amazon.com/dp/0521401534/?tag=pfamazon01-20

I must say though, if you want to get to the heights that Zemanian did then the other book I listed you're going to probably have to read too, or get that level of information else where.

The control theorist who was in my analysis class said his advisor for his ph.d use to be a pure mathematician. The reason he was taking the class, was because he needed to be able to do really rigorous proof. So depending on what you do in life, it is possible outside of mathematics to have to be able to do really rigorous mathematics.

I'm glad I could help.
Best of luck!

 

[NathanExplosion]

No problem at all. You're definitely going to like it. I feel akin with EE types. I was an electronic technician in the Navy and had received 2 years of electronics training. I know EE uses a ton of math. My electronics professor in college worked on some really mathematical intense stuff.

Here's a book he wrote that's at amazon. Impressive the level of mathematics he used.
https://www.amazon.com/dp/0521401534/?tag=pfamazon01-20

I must say though, if you want to get to the heights that Zemanian did then the other book I listed you're going to probably have to read too, or get that level of information else where.

The control theorist who was in my analysis class said his advisor for his ph.d use to be a pure mathematician. The reason he was taking the class, was because he needed to be able to do really rigorous proof. So depending on what you do in life, it is possible outside of mathematics to have to be able to do really rigorous mathematics.

I'm glad I could help.
Best of luck!

Zemanian was your prof? I'm assuming you took ESE 271 (Circuit Analysis I) at Stony Brook then?

 

[math_owen]

Indeed that's the man.