Questions about papers on Mathematical Physics

[Jimmy84]

Hi everyone
Im going to start a major in physics next year and I would like to follow mathematical physics afterwards but I lack knowledge about which are the areas of math that contribute the most to the development of physics. For instance I know that differential geometry, functional analysis and abstract algebra specially group theory have deep implications for physics.

I would like to be prepared to write papers and to learn on my own beforehand the most applicable branches of math without having all the knowledge and advantages that a phD students has.

Which books are the best ones to learn mathematical physics?

Which mathematical background do I need in order to learn C algebras?

Which are the areas of math that are the most useful to do research for mathematical physics?

Thanks a lot in advance.

 

[VeeEight]

There are lots of areas of math to study for physics. Some are Functional Analysis, Topology, some Measure Theory, Complex Analysis, Representation Theory, Group Theory and Differential Geometry. The actual uses of these topics becomes apparent when you study more and need to develop more tools.

 

[Kreizhn]

It really depends on what area of physics you plan on entering, as the mathematical aspects vary widely depending on the field. Even pure-mathematical concepts that might not seem like they can be applied to physics have found ways to creep into the study.

For example, it seems unlikely to me that there are many areas of physics that would require an in-depth knowledge of category theory, but I do have a friend working in Quantum Foundations that uses it a fair bit. I think module theory is used in chemical physics but not widely, and hence probably wouldn't be worth studying (other than possibly the repercussions of representation theory).

I cannot speak for things like condensed matter physics and the like, but if you plan on going into relativity theory or quantum mechanics a good short list would be functional analysis, group theory, and differential geometry.

 

[Jimmy84]

It really depends on what area of physics you plan on entering, as the mathematical aspects vary widely depending on the field. Even pure-mathematical concepts that might not seem like they can be applied to physics have found ways to creep into the study.

For example, it seems unlikely to me that there are many areas of physics that would require an in-depth knowledge of category theory, but I do have a friend working in Quantum Foundations that uses it a fair bit. I think module theory is used in chemical physics but not widely, and hence probably wouldn't be worth studying (other than possibly the repercussions of representation theory).

I cannot speak for things like condensed matter physics and the like, but if you plan on going into relativity theory or quantum mechanics a good short list would be functional analysis, group theory, and differential geometry.

Thanks, I am looking perhaps for a field of math that might not be the most difficult or technical but that could allow myself to be able to write papers about it at some point.
I intend to learn on my own some differential and algebraic geometry. But lately I have been considering learning more about functional analysis and measure theory.
Im also interested in the foundations of physics.
What do you guys think about that, is it a realistic enterprise to axiomatize physics ?

Another intersting area of research is constructive quantum field theory. But I can imagine it might be very difficult to find a job in that area.
What is the mathematical background needed in order to figure out that problem?
and does anyone knows good books about mathematical physics to start with?
Thanks.

 

[CellCoree]

As a scientist, it's great to see your enthusiasm for mathematical physics and your desire to prepare yourself for future research in this field. Mathematical physics is a highly interdisciplinary field that combines concepts from both mathematics and physics to understand the laws of nature and their mathematical representations. It is a rapidly evolving field, and there are constantly new developments and applications being made.

To answer your questions, there are several areas of mathematics that are crucial for the development of mathematical physics. As you mentioned, differential geometry, functional analysis, and abstract algebra are all important branches of math in this field. In addition, topology, complex analysis, and probability theory are also highly relevant. It is important to have a strong foundation in all of these areas to fully understand the mathematical concepts used in physics.

In terms of books, there are many great resources available for learning mathematical physics. Some popular textbooks include "Mathematical Methods for Physicists" by George Arfken and Hans Weber, "Mathematical Methods in the Physical Sciences" by Mary L. Boas, and "Mathematical Methods of Classical Mechanics" by V.I. Arnold. It's also helpful to read research papers in the field to gain a deeper understanding of current topics and techniques.

To learn about C algebras, it's important to have a strong background in linear algebra, functional analysis, and topology. A good introductory book on the subject is "C*-Algebras and Operator Theory" by Gerard J. Murphy. It's also helpful to have some knowledge of quantum mechanics and group theory.

In terms of research, the most useful areas of math for mathematical physics can vary depending on the specific topic you are interested in. However, some general areas that are commonly used in research include differential geometry, topology, and functional analysis. It's also important to have a good understanding of mathematical modeling and numerical methods.

Overall, my advice would be to have a strong foundation in all of the relevant areas of mathematics and to continuously expand your knowledge by reading textbooks and research papers. Additionally, don't be afraid to reach out to experts in the field for guidance and advice. Best of luck in your studies and future research endeavors!