# What type of book is best for QM and GR - math or physics?

I decided to go deep in physics theories such as QFT, general QM and Special/General Relativity. Would it be better to spend a lot of time, say, 1+ year, learning through the most complete math books or just use books that mix math and physics to learn the necessary and suficient math and go to the physics?

vanhees71
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2021 Award
I've also been self-learning math and physics when I was in highschool. Usually I just started with a subject I was interested in and then realized what mathematical tools I needed. This is, however, indeed not the best way to learn physics. It's way better to go systematically in the "canonical order", i.e., Newtonian mechanics, then classical electrodynamics including special relativity. Then you may decide, whether to first go on with GR or quantum mechanics. For quantum mechanics also you need to get a good grip on the non-relativistic theory. QFT is more subtle.

Concerning the math you can do this in parallel, because a lot of concepts of math have their origin in physics anyway (it's not by chance that Newton discovered calculus when thinking about mechanics; on the other hand Leibniz discovered it by pure mathematical interest). In classical mechanics you need first of all Euclidean vectors, derivatives, integrals, and then some ordinary differential equations. Mathwise, I think the most challenging step is to learn the full 3D Euclidean vector calculus with div, grad, curl, as well as line, surface, and volume integrals and their various interrelations (Gauss's and Stokes's Theorems), which you need in full glory for classical electrodynamics. At the same time electrodynamics is the best subject to learn its use, together with the most important physically relevant partial differential equations, since a loarge part of classical electrodynamics is mathematically a linear field theory.

Concerning special relativity, I think the optimal order is to split it in two parts. You can already learn a part of special relativity after learning Newtonian mechanics, introducing 4D vector algebra (Minkowski space) on the same level as 3D Euclidean vector algebra. The second part then comes into the game when you have learnt classical electromagnetism in the usual 3D vector-calculus way. After this you can extend your knowledge about 4D Minkowski space to full 4D vector calculus.

Before dealing with quantum mechanics, I'd also recommend to learn about "the action principle" and together with it calculus of variations and some elementary (Lie) group theory, because that's the best way to understand the heuristics of quantum mechanics.

kent davidge
PeroK
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I decided to go deep in physics theories such as QFT, general QM and Special/General Relativity. Would it be better to spend a lot of time, say, 1+ year, learning through the most complete math books or just use books that mix math and physics to learn the necessary and suficient math and go to the physics?

I wouldn't spend too long on just the mathematics. But, if you take a look at "Paul's online maths", I'd say you need everything in his Calculus courses. You may be able to learn it from there, or you may need some textbooks.

For QM you also need a good introduction to Linear Algebra and Complex Numbers.

As others have said, there are physics prerequisites as well. Classical mechanics, Lagrangian and Hamiltonian mechanics, EM (field theory).

But also, it's experience and familiarity with the various physical and mathematical concepts.

kent davidge and vanhees71