Mathematics path for physics?

[Devin]

Hi, I wish to not only learn, but prove every theory I come across. This requires a ton of math research, and at this point, I am about to begin quantum mechanics, and general relativity after I finish up my differential geometry book. My question, I suppose, is after I finish differential geometry, and tensor calculus (assuming I've met all the prereqs for it), what mathematics and physics should I learn before I do 1) Lorentz force derivation, 2) Quantum (chromodynamics and electrodynamics)

 

[rootone]

I admire your enthusiasm.
Your next step is whatever you think is the most interesting.

 

[DrClaude]

Hi, I wish to not only learn, but prove every theory I come across.

What does that mean? How do you "prove" a physical theory?

 

[Devin]

What does that mean? How do you "prove" a physical theory?

I thought it to be implied that proof, and or derivation of any of the mathematical constructs involved, and for that, I apologize.

 

[Devin]

I admire your enthusiasm.
Your next step is whatever you think is the most interesting.

thank you :)

 

[bigfooted]

Gerard 't Hooft (Nobel prize physics 1999) has a long list of topics he thinks you should study if you want to become a good theoretical physicist. He also gives some subtopics and links to online lecture notes.

http://www.staff.science.uu.nl/~gadda001/goodtheorist/

My opinion is that physics and mathematics knowledge should be acquired following a pyramid structure: you need a lot of general physics and mathematics (calculus, differential equations, classical mechanics) before you can move to more advanced topics. If your path toward specialist knowledge is too narrow, your fundamental understanding in certain related fields is too weak and you will not be able to fully comprehend/appreciate the theory and you will certainly not be able to contribute to the field.

 

[bcrowell]

Hi, I wish to not only learn, but prove every theory I come across. This requires a ton of math research, and at this point, I am about to begin quantum mechanics, and general relativity after I finish up my differential geometry book. My question, I suppose, is after I finish differential geometry, and tensor calculus (assuming I've met all the prereqs for it), what mathematics and physics should I learn before I do 1) Lorentz force derivation, 2) Quantum (chromodynamics and electrodynamics)

None of the fancy math you describe is necessary in order to derive the Lorentz force. (Actually, it's ambiguous to say that you want to "derive the Lorentz force." It would have to be derived from some assumptions that you consider more fundamental.)

 

[radium]

Depends what you mean by prove. A physicists proof and a mathematicians proof are much different. It also depends on how far you want to go back to derive something. You could derive the Lorentz force by using the special relativity you generally learn freshman or sophomore year, you could use the covariant formalism of EM. Or you could even go way back to first principles and basically rederive EM by seeing where the EM field comes from just using symmetry, find the action get the equations of motion, identify conserved quantities, etc. You could even later generalize this to Yang Mills theory, add matter fields, quantize it, etc. Maybe you could even generalize to any dimension (QED is very interesting in 2+1d.

In order to do the latter you would need to know about Lie groups, some differential geometry, and a lot of other more basic things.