Importance of pure math books for aspiring physicists

[Thinker301]

Hi everybody!

I was wondering how much importance there is in studying pure math, for the budding theoretical physicist? mathematical physicist?

Also I was wondering whether a mathematical physicist would work more on mathematical methods that physicists can use or do work like Einstein or Feynman?

Thank you for your time!

With my heart on my sleeve,
Thinker301

 

[WannabeNewton]

"Theoretical physicist" can mean a lot of different things and each subfield of theoretical physics requires a different set of mathematical tools and varying degrees of mathematical rigor. Without more specificity such a question is impossible to answer. What kind of physics are you talking about? Condensed matter theory? AMO? HEP theory? Gravitational physics? "Theoretical physics" isn't a subfield in and of itself. That's like asking what kind of advanced lab is most useful for "experimental physics".

Are you asking if a physics undergrad should devote precious time to proper studying of pure math for it's own sake?

Mathematical physics is basically a subset of pure math so that question unequivocally answers itself. I'm not sure what you mean by "Also I was wondering whether a mathematical physicist would work more on mathematical methods that physicists can use or do work like Einstein or Feynman?". Are you asking if mathematical physicists would be doing the things that Feynman or Einstein did? If so then no they definitely do not. You should look up descriptions of mathematical physics from various universities to get a better idea of what it actually is. It's much more aligned with pure math than it is with physics. I don't know about Feynman but the stuff that Einstein did was far from mathematically rigorous. He relied more on ingenious physical insight than he did rigorous and logical mathematical arguments.

 

[Thinker301]

My bad. Say I were to specialize in String Theory or Gravitational physics, would pure math help me understand the math better, and would that help me understand the physics better?

 

[TheAustrian]

To really learn Physics, you need to do three things:
- Study Maths
- Study Maths
- repeat all of the above

 

[micromass]

To really learn Physics, you need to do three things:
- Study Maths
- Study Maths
- repeat all of the above

Sure, math is really important in physics. But it really matters what kind of math you study. Something abstract ring theory seems not very useful in physics and you're better off studying something else.

Also, physics is not math. The skills you need to solve physics problems are very different than what you need to do mathematics. You need a physical intuition. You won't get this from studying pure mathematics.

That said, if the OP is interested in string theory, then he is going to need a lot of pure mathematics. It is one of the only branches in physics that really requires a lot of pure mathematics. So yes, I would study pure math books in the case of the OP. In the case of many other students, I would not study pure math so much.

 

[TheAustrian]

Sure, math is really important in physics. But it really matters what kind of math you study. Something abstract ring theory seems not very useful in physics and you're better off studying something else.

Also, physics is not math. The skills you need to solve physics problems are very different than what you need to do mathematics. You need a physical intuition. You won't get this from studying pure mathematics.

That said, if the OP is interested in string theory, then he is going to need a lot of pure mathematics. It is one of the only branches in physics that really requires a lot of pure mathematics. So yes, I would study pure math books in the case of the OP. In the case of many other students, I would not study pure math so much.

Agreed, but that is why approx 40% of a Physics UG course are math classes. In the 4th year of an UG class, Physics problems become more mathematics than Physics, and it is entirely possible to pass most classes using just Maths knowledge.

 

[WannabeNewton]

My bad. Say I were to specialize in String Theory or Gravitational physics, would pure math help me understand the math better, and would that help me understand the physics better?

I can't comment on string theory because I don't know anything about it nor do I have even the slightest inkling of interest in it. However I can comment on gravitational physics if restricted to classical general relativity, modern cosmology, stellar evolution, and related fields. In this case, based on what I've been told by multiple professors as well as from personal experience, a proper study of pure math really wouldn't help you. Mind you it wouldn't necessarily hurt you but it won't help you in any noticeable way either.

At the pedagogical level, that is at the level where you are formally learning e.g. general relativity, the math (say differential geometry) presented at the level of the standard GR books will be more than enough if your intent is to study the physics of GR (as opposed to the mathematical physics such as stability of solutions to Einstein's equations for example). You aren't really going to understand the physics any better if you know differential geometry at the level of Spivak or Kobayashi/Nomiru (pure math books on diff geo) as opposed to just knowing it at the level of Wald or Straumann (GR books). That being said, a proper study of pure math does let you acquire very useful problem solving skills and modes of thinking that may prove useful in physics.

In the 4th year of an UG class, Physics problems become more mathematics than Physics, and it is entirely possible to pass most classes using just Maths knowledge.

This is entirely incorrect. I can't even fathom where you got such an idea.

 

[micromass]

Agreed, but that is why approx 40% of a Physics UG course are math classes. In the 4th year of an UG class, Physics problems become more mathematics than Physics, and it is entirely possible to pass most classes using just Maths knowledge.

Yes, but there is a big difference between being able to pass classes and knowing physics. There might be some correlation, but I don't consider the two to be the same thing.

 

[Thinker301]

Thanks everyone! So say I were to pursue string theory. What math would I need to begin research?

I know there is :
Calc
Differential equations
Linear algebra
Real analysis
Complex analysis
Topology
Differential geometry
Abstract algebra

 

[TheAustrian]

Your list is a good start.

 

[Arsenic&Lace]

Newton, your statement that pure mathematics courses can provide useful problem solving skills, while true, must also be held to the following caveat: That mathematicians will think about mathematical structures in a very different way from physicists, and that for a researcher in physics, this perspective appears to be more damaging than useful.

Consider, as a case in point, the difference between how a course in real analysis views series to how a course in quantum field theory views series.

One must bear in mind the dramatic distinction between these views.

 

[lurflurf]

Consider, as a case in point, the difference between how a course in real analysis views series to how a course in quantum field theory views series.

Interesting, care to elaborate?

 

[Arsenic&Lace]

Disclaimer: I'm no pro at field theory or analysis, so don't take any of this at face value. Hopefully more informed members of the boards will spot any egregious errors.

How much elaboration would you like? My understanding is that the main interest of mathematicians involves studying the general properties of series and looking for rules which can be used to determine if a given series will converge (I took an intro course in real analysis a very long time ago). I would be surprised if an analysis course even mentioned using the series in an engineering or physics problem. They are not interested in specific series, but rules which govern all series.

A physicist has a specific series, and what is more, his/her series from field theory diverge at higher orders with loops. So the problem is, given this mathematically undesirable series, what physical/mathematical scheme could possibly scuttle the infinities and provide one with a sensible result? The scheme is called renormalization.

I don't know if a rigorous, consistent mathematical framework has ever been derived for this methodology, and it would probably seem dubious to the typical mathematician, but if the typical mathematician is reading this, s/he should feel free to contradict me if so inclined.