What advanced math should I study to become a successful physicist?

[Jorge G.]

I know that is a vague question, but here me out.

I plan on getting my PhD in physics soon.

I've been told several times that all you need to become a physicist is Calculus I-III, differential equations, linear algebra, and maybe vector analysis and advanced differential equations, and the rest of the math you need, you will be taught in your physics courses.

While that has held true, recently I have heard two things that have made me think. The first was the quote, "physics is too hard for physicists." which I'm most of you've heard.

The second was something about Einstein intensely studying differential geometry before developing general relativity.

This made me think that I, as a theoretical physicist, wanting to reach the ranks of Einstein, must endeavor deep into the mathematical realm.

Therefore I ask what fields of mathematics should I study to be more than successful as a physicist?

 

[blaughli]

I'm in a class called "Theoretical Physics" right now, and we are basically just being introduced to new notation (much of which Einstein himself invented to keep track of all of the indices that you deal with when you're doing calculations in 3D space - check out the Einstein summation convention, other Einstein conventions...). We are also being re-introduced to the key concepts for physicists from Linear Algebra, and I suspect that later this semester we'll be re-introduced to the key DiffEq stuff. Part of the benefit of this class are the specific applications of things we should already have learned to some standard physics problems, such as the normal modes of masses on springs and pendulums.

There is also a graduate level version of this class, which I plan to take later. The graduate version is a pre-req for my school's General Relativity class. Sortof makes sense that you need to learn Einstein's conventions before you can dive into Einstein's theories, right?

So I suggest the normal route. Linear Algebra is huge. Calc I-III is a must. DiffEq's allow physics problems involving change to be solved, so got to know it well. Take theoretical as soon as you've done that stuff.

Just my two cents - I'm a beginner in physics. Like you, I have realized that math is the language and you want to be fluent before you travel to Physics Land!

 

[espen180]

I guess it depends on what kind of physics you are studying, but in general you need a boatload of math, much more than a mathematician would need. I can name some topics which are used in theoretical physics and in research in high energy physics / quantum gravity / string theory.
It is impossible to learn all of this during one Phd, so you should learn what you need as you need it. These are just the areas of mathematics you should be aware of, and you should know what each one is about so that you can recognize what you need and when.

- Topology (Point-set, differential and algebraic topology)
- Differential geometry and analysis on manifolds (In the abstract manifold setting)
- Functional analysis
- Complex analysis (Complex manifolds, Kähler and Calabi-Yau manifolds)
- Abstract algebra (Groups, rings, algebras, modules, categories ...)
- Lie Groups and Lie algebras
- Representation theory of groups, rings and algebras
- Gauge theory
And if you plan on doing topologial quantum field theory,
- Higher category theory

To get a jist of what is going on in current research, you should browse the Arxiv regularly.
http://arxiv.org/

 

[Jorge G.]

Thank you both for the input.

 

[chill_factor]

I should add that you need that math if you're going into theory. In experiment you don't need that sort of math. Seriously you don't. Just read some experiment (non-high energy) papers in Arxiv or in physical review. There's not much math in there.

However you still need some pretty high level math just to pass the grad classes, and there's stuff that just isn't taught...

 

[homeomorphic]

Math can add a lot to your understanding of physics. It's not clear to me what's essential and what's not. In special relativity, there are a couple of nice things from math. There is some very interesting hyperbolic geometry lurking in Minkowski spacetime. Lorentz transformations are really Mobius transformations from complex analysis in disguise. Many Mobuis transformation proofs turn out to be more elegant than the space-time ones.

Quaternions and Clifford algebras provide some key ingredients to understanding particles like the electron.

I was playing around with stereographic projection and the Hopf fibration to better understand 2-dimensional state spaces, such as that of the electron spin. I derived the Pauli matrices this way. It was cool. I have to write it down sometime. More or less original on my part, though I am sure it's not completely new to people in the know.

I think knowing a little geometry and topology sheds a lot of light on classical mechanics (see Arnold's classical mechanics book). Differential forms shed light on Hamiltonian mechanics and electromagnetism.

Quantum mechanics and QFT are surprisingly combinatorial. Understanding things like generating functions from combinatorics (with a little category theory) provides a lot of insight, as Baez and Dolan have shown.

It's nice to know about Lie algebras in quantum mechanics. The symmetries of nature are often Lie groups, and Lie algebras are the "small elements" of Lie groups. If you talk about things like infinitesimal rotations, those are really Lie algebra elements. This plays a role in the theory of angular momentum in QM, for example.

Just some examples. I think if you want a deep understanding, you need quite a bit of math.

 

[chill_factor]

How do you know all that stuff? I feel like its hard even to study the physics alone, much less all the math behind it rigorously.

 

[homeomorphic]

Well, I don't know the physics side that well. Took a long time. It's probably why my thesis isn't done yet after 6 years of grad school.