Mathematics book for a Physicist

[JK423]

I'm a graduate student of theoretical physics.
I'd like to specialize in the field of quantum information, quantum theory in general. For that reason, i`d really like to study the theory of Hillbert space, or/and operators, for example, in mathematical detail so that i'll be able to have a better understanding.
Why I'm saying this:
We, physicists, usually don't have a solid mathematical background. We use mathematics quite sloppy. One example is the misuse of operators in quantum mechanics ending in paradox situations, just because we don't know about the mathematical mistakes that we do without knowing. I want to fix that for myself.
So, i would really like you to suggest books on the theory of Hillbert space and operators. I'm not used to the mathematical language, so if i have to learn first some things on the mathematical language and logic then please suggest (very introductory) books on that as well.

Thanks a lot

John

 

[issacnewton]

Hi John

If you want to become a theorist, I would suggest taking math courses, starting with
Real Analysis. Then you can go to Functional Analysis. I think that Hilbert spaces come
under Functional Analysis. Since you want to study Quantum information theory, I would say, you need to study mathematical logic and abstract algebra. You are right about sloppy math of physics community. Its sad. Good theorist should be a good mathematician, I say.

Physicists need to learn little rigor from mathematicians...

 

[JK423]

Thanks for your reply.
I think i have studied enough Real Analysis as an undergraduate. So in order to understand the Hillbert space, i should study Functional Analysis.
About quantum information, why would you suggest abstract algebra and not Functional Analysis? Since a Hillbert space is involved there as well.
And one last thing. Do you know any good -very introductory/readable- books on Functional Analysis and/or Abstract Algebra?

 

[issacnewton]

Well, I had read some articles which stressed imporatance of abstract algebra in that area. I also did study Real Analysis from Witold Kosmala's book. But I have never studied
abstract algebra, though I want to.

About the books, I don't know much. If you find good book on Abstract Algebra, let me know... since I want to learn it too.

If you already know a bit of analysis, then there is good book for learning counter examples in analysis ."Counterexamples in Analysis (Dover Books on Mathematics)"

The math education in physics departments is pathetic in my opinion. Physics students
are ill prepared for the kind of mathematics in use in many areas of theoretical physics

 

[micromass]

Here are some great books that I really recommend:

For functional analysis, you should read "Introductory functional analysis" by Kreyszig. The good thing about Kreyszig is that he explains the math in a very good way, without assuming to much knowledge of the student. If I could choose a book to start in functional analysis, then it would be this one!

For abstract algebra, a great book is "a book on abstract algebra" by Pinter. It's one of the best math books I've read. It reads more like a novel than a math book, and it contains everything a first course should contain. (I do recommend to make lots of exercises, because he often states important things there. So you should at least read all the exercises...)

These are two of the best introductory math books that I've encountered. However, don't expect to much advanced material in there. Once you've gotten acquainted with the material, you'll need a more advanced book to fill in the gaps...

 

[JK423]

Thank you very much Micromass! I'll check these books.
I heard that Functional Analysis books assume that you know 'topology', is that true for Kreyszig?

 

[micromass]

Thank you very much Micromass! I'll check these books.
I heard that Functional Analysis books assume that you know 'topology', is that true for Kreyszig?

No, that's why I recommended Kreyszig. He assumes no knowledge of topology of any kind. Of course, he will work with metric spaces and stuff. But if you haven't seen that, then there's an introduction to these in his books.

The only math you'll need in Kreyszig is linear algebra and calculus. The more knowledge you have, the better of course...

 

[JK423]

That's just great..
One last question please, what should i start reading first, abstract algebra or functional analysis? Or are they independent..?

 

[micromass]

That's just great..
One last question please, what should i start reading first, abstract algebra or functional analysis? Or are they independent..?

They are independent. Of course, you'll need to know linear algebra for both. But if you're comfortable with linear algebra, then you could read both easily.

Personally, I would read the functional analysis first, since there are more links with physics in functional analysis. If you read abstract algebra, then you won't see the connection with physics very easily, so that could work very demotivating...

 

[MannyP2011]

I'm a graduate student of theoretical physics.

Hey John, my name is Manny and I am entering my junior year this coming fall and I am planning on becoming a theoretical physicist. I would really appreciate it if you would give me an idea on the mathematics classes that I should take. Should i start with Real Analysis, and is there anything you recommend ?

thanks manny

 

[JK423]

Hi Manny!
Since you know from the beginning that you want to become a theoretical physicist, you should take as many mathematics classes as you can.
Here in Greece we didn't have the option to choose what classes to attend, we had the following obligatory math classes:

Linear Algebra
Real Analysis 1, 2
Complex analysis
Probabilities-Statistics
Ordinary differential equations
Partial differential equations
Group theory (optional)

You are going to need everything (and ofcourse more) in the future, so you either study these early on, or you'll be obliged to study them by yourself later on.


John

 

[MannyP2011]

thank you i really appreciate it ... if you have any other info i am open .. thanks again

 

[dextercioby]

I'm a graduate student of theoretical physics.
I'd like to specialize in the field of quantum information, quantum theory in general. For that reason, i`d really like to study the theory of Hillbert space, or/and operators, for example, in mathematical detail so that i'll be able to have a better understanding.
Why I'm saying this:
We, physicists, usually don't have a solid mathematical background. We use mathematics quite sloppy. One example is the misuse of operators in quantum mechanics ending in paradox situations, just because we don't know about the mathematical mistakes that we do without knowing. I want to fix that for myself.
So, i would really like you to suggest books on the theory of Hillbert space and operators. I'm not used to the mathematical language, so if i have to learn first some things on the mathematical language and logic then please suggest (very introductory) books on that as well.

Thanks a lot

John

For the interest in quantum information you don't need to know functional analysis, just to work well with the Dirac formalism in which case you need the math at the level of Sakurai's <Modern Quantum Mechanics> text: linear algebra and differential equations. However, if, for your own interest, you want to understand the mathematical background and foundation of the commonly accepted theory of QM, you need to know functional analysis, group theory and the mixture of the 2, the latter brilliantly described in Barut & Raczka's text on group theory.

 

[A. Neumaier]

I'm a graduate student of theoretical physics.
I'd like to specialize in the field of quantum information, quantum theory in general. For that reason, i`d really like to study the theory of Hillbert space, or/and operators, for example, in mathematical detail so that i'll be able to have a better understanding.

Try the online book
Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras, 2008.
http://lanl.arxiv.org/abs/0810.1019

 

[George Jones]

For functional analysis, you should read "Introductory functional analysis" by Kreyszig. The good thing about Kreyszig is that he explains the math in a very good way, without assuming to much knowledge of the student. If I could choose a book to start in functional analysis, then it would be this one!

I really like this book as well. It was used as the text for an undergrad functional analysis course that I took.