# Books teaching maths more advanced than what is used in quantum physics

• Quantum

## Main Question or Discussion Point

Hello friends.

I was thinking that if you study physics, you learn first the basic math to understand it, but you don't go beyond, or deeper, you just learn the minimum math necessary to understand what you are going to see in your physics leassons.

For example, if you are going to learn quantum mechanics, you first learn about linear algebra, Hilbert Spaces... but you don't learn beyond that, in this particular example I would like you to tell me that I could learn about Banach Spaces, Functional Analysis etc.

I was wondering if you can recommend me books, or tell me about the discipline inside mathematics that is more advanced than the used in physics.

There are several formulations of quantum mechanics, you have differential equations, matrices, you have path integral, gauge theory, abelian gauge theory,

In other words, I would like to learn math a lot more advanced than the used in physics, Is there any book, blog post or thread talking about it?

## Answers and Replies

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The board is not allowing me to quote as completely as it should so I've edited away my reply.

The board is not allowing me to quote as completely as it should so I've edited away my reply.

Instead, I suggest that you check out Professor Terrence Tao -- he's great.

dextercioby and sysprog
jim mcnamara
Mentor
Just a helpful comment, based on the way you put together your question.
If you have a math degree already, it is possible that taking on an advanced book for learning something new will work.

Otherwise if you just want to get an idea of where you stand on the Mount Everest of Mathematical Studies - there are LOTS of them - try going to mathoverflow, just look at what they are dealing with.
https://mathoverflow.net/

Hey I went there to get the link for you - Cool, there's an article on epidemiologic modeling of Coronavirus.

jonjacson and sysprog
jim mcnamara
Mentor
Also - https://polymathprojects.org/

Leading mathematicians get together and pound away on problems.
They create project + sites with URLS like Polymath13. The link I gave you gives a lead into Polymath16, for example.

jonjacson and sysprog
This book could be a great source of advanced mathematical training for theoretical physics.
This one by Roger Penrose could provide a broader picture of the kind of mathematics being used in different theoretical formulations of quantum as well as the general relativity and beyond.

Keith_McClary
Extensive "preview" :GEOMETRY, TOPOLOGY AND PHYSICS MIKIO NAKAHARA.
yes, I referred to this book for the unitary operator mathematical structure in QM. That's why i suggested. As an undergrad i could understand it after going through the Feynman lectures III. I cannot however vouch for the rest of the topics covered as to how well they are covered and explained. I am still a physics baby!

George Jones
Staff Emeritus
Gold Member
Extensive "preview" :GEOMETRY, TOPOLOGY AND PHYSICS MIKIO NAKAHARA.
yes, I referred to this book for the unitary operator mathematical structure in QM. That's why i suggested. As an undergrad i could understand it after going through the Feynman lectures III. I cannot however vouch for the rest of the topics covered as to how well they are covered and explained. I am still a physics baby!
Nakahara is a standard, but I want to mention several other references that are personal favourites for me.

A book worth looking at is "Differential Geometry and Lie Groups for Physicists" by Marian Fecko. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge field theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,
A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).
I have found that this format works well for me, but other folks might have different opinions

Fecko is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf

Also, the book A" Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry" by Peter Szekeres is nice. This book does not cover as much differential geometry as Nakahara, but what it does cover, it treats more rigourously.

Finally, a book that will not be eveyone's cup of tea. The book "Mathematical Physics" by Robert Geroch, which is in the orthogonal complement to the set of most books with similar titles, starts with a brief introduction to category theory. This very nice broad introduction to some abstract pure maths contains a broad survey of abstract algebra, topology, and functional analysis, and it does a wonderful job at motivating (mathematically) mathematical definitions and constructions. However, its layout is abominable. Surprisingly, since Geroch is an expert, it contains no differential geometry. This book, which requires real effort, and which contains only a few applications to physics, is a book on which I spent a fair bit of time 25 or 30 years ago.

dextercioby, atyy, vanhees71 and 2 others
vanhees71
Gold Member
2019 Award
What do you think about Sadri Hassani, Mathematical Physics? I kind of like it as a reference, but it's also a bit on the dry side of Bourbaki style of math-textbook writing, which is pretty hard for physicists who need some intuition behind the mathematical methods, but as a reference source for mathematical definitions and theorems it's good.

atyy
A book worth looking at is "Differential Geometry and Lie Groups for Physicists" by Marian Fecko.
I like Fecko very much also.

vanhees71
Math_QED
Homework Helper
2019 Award
So you are looking for a functional analysis book. Great, but what functional analysis you want to know more about? Functional analysis is a huge field nowadays.

vanhees71
robphy
Homework Helper
Gold Member
Finally, a book that will not be eveyone's cup of tea. The book "Mathematical Physics" by Robert Geroch, which is in the orthogonal complement to the set of most books with similar titles, starts with a brief introduction to category theory. This very nice broad introduction to some abstract pure maths contains a broad survey of abstract algebra, topology, and functional analysis, and it does a wonderful job at motivating (mathematically) mathematical definitions and constructions. However, its layout is abominable. Surprisingly, since Geroch is an expert, it contains no differential geometry. This book, which requires real effort, and which contains only a few applications to physics, is a book on which I spent a fair bit of time 25 or 30 years ago.
You may be interested in some lecture notes recently published by the Minkowski Institute Press
http://www.minkowskiinstitute.org/mip/books/ln.html

Geroch, General Relativity: 1972 Lecture Notes
Geroch, Quantum Field Theory: 1971 Lecture Notes
Geroch, Geometrical Quantum Mechanics: 1974 Lecture Notes
Geroch, Unsolvable Problems: 1990 Lecture Notes
Geroch, Differential Geometry: 1972 Lecture Notes
Geroch, Topology: 1978 Lecture Notes
Geroch, Infinite-Dimensional Manifolds: 1975 Lecture Notes

robphy
Homework Helper
Gold Member
@robphy

This is the link to UoCh for Robert Geroch's Lecture Notes. I think/hope its legal to download the pdfs.
That’s from Bob Geroch’s personal webpage.
He made available the $\LaTeX$ed drafts of the published notes.
Additional notes (not necessarily $\LaTeX$ed) are linked from the parent directory https://home.uchicago.edu/~geroch/ )

George Jones
Staff Emeritus
Gold Member
What do you think about Sadri Hassani, Mathematical Physics? I kind of like it as a reference, but it's also a bit on the dry side of Bourbaki style of math-textbook writing, which is pretty hard for physicists who need some intuition behind the mathematical methods, but as a reference source for mathematical definitions and theorems it's good.
I like Mathematical Physics by Hassani, particularly the second edition, which, for example, has material on fibre bundles and gauge theory that is not in the first edition. Hassani's biographical sketch of the university drop-out Weierstrass is not dry, as it includes: "He infuriated his father by rarely attending lectures, getting poor grades, and instead becoming a champion beer drinker. He did manage to become a superb fencer, but when he returned home, he had no degee." Sounds like Weierstrass would fit it with some of today's university students.

atyy and vanhees71
Sounds like Weierstrass would fit in with some of today's university students.
When was the last time the older generation had faith in the younger ones?

atyy
vanhees71
Gold Member
2019 Award
Weierstraß was anyway a funny character. I once read an anecdote about him, when he was a school teacher. One day, the director of the school was very angry since Weierstraß missed to come to teach his math class one morning. When the director came to Weierstraß's house, he found them deep in thought over a pile of calculations concerning the elliptic functions ;-)).

atyy
FactChecker
Gold Member
In other words, I would like to learn math a lot more advanced than the used in physics, Is there any book, blog post or thread talking about it?
Math is such a tremendously diverse subject that you probably should limit or describe the direction that you want to head in. For instance, are you interested in geometrical-type mathematics or in abstract mathematics, or in number theory, etc. The mathematics that is used in physics is as advanced as any other mathematics, but there are some subjects (like category theory, Galois theory) that are very different IMO.

Math is such a tremendously diverse subject that you probably should limit or describe the direction that you want to head in. For instance, are you interested in geometrical-type mathematics or in abstract mathematics, or in number theory, etc. The mathematics that is used in physics is as advanced as any other mathematics, but there are some subjects (like category theory, Galois theory) that are very different IMO.
What is the roadmap to the Geometrical - type mathematics learning? Could you please give some guidelines?

FactChecker
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What is the roadmap to the Geometrical - type mathematics learning? Could you please give some guidelines?
Sorry. I may have opened a can of worms that I am not qualified to answer. In the last few years, I have been interested in geometric algebra and geometric calculus, but I am old and it has a steep (IMHO) learning curve.

Sorry. I may have opened a can of worms that I am not qualified to answer. In the last few years, I have been interested in geometric algebra and geometric calculus, but I am old and it has a steep (IMHO) learning curve.
I just googled the terms geometric algebra and calculus. I gather that these things are not taught as a part of the usual undergrad curriculum, perhaps. A nice historical tree of the subject development.

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