I would like to learn/brush up my concepts on Mathematics related to physics and engineering. I have found this book (Mathematical Methods for Physicists, by Arfken, Weber, Harris). This book covers almost all the topics that I need, However, it doesn't explain how and why things happen like this book (Elementary Differential Equations and Boundary Value Problems,by Boyce and Di Prima) explains the differential equations. Need something like the latter book to understand the mathematical concepts related to physics and engineering that would explain things elaborately.
[Math] Engineering Mathematics and Mathematical Physics Books
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This got too long for a comment but is meant to be an extended one. I'm note quite the guy to say being mostly interested in it from a structural/mathematical perspective. Forgive me if I'm not telling you anything new.
You can definitely do TQFT within the confines of pure math. If what you want is the standard model you'll do well to understand your representation theory, as types of particles correspond to fundamental representations of Lie groups ($U(1)\times SU(2)\times SU(3)$ in the standard model, times the Poincaré group if you do the analysis.) From there a quantum field is a section of a vector bundle associated to the representation over space-time satisfying a variational principle (an extremal of an action) involving suitably equivariant connections (which are incidentally your bosons). Faria-Melo develops this and in fact exhibits the standard model in this framework.
They leave out a clear analysis of how representations tie in with types of particles, but this is done by Baez and Huerta in this text (http://math.ucr.edu/~huerta/guts/). Basically, elements in your fundamental representations are fermionic particle states, generators of the adjoint representation are bosons that act on your fermions in a way that can be represented by Feynmann diagrams.
Quantization is still fluffy to me, but it appears this is where quantum groups come in: You cannot deform a semi-simple Lie algebra and get a reasonable deformation of its representation theory (it's category of representations). You can however deform its universal enveloping algebra (which is a Hopf algebra, i.e., an object with a favourably interacting product and coproduct). There is a master class on this going on right now which talks about this for the purpose of studying 3-manifold invariants using 3-dimensional field theories. Notes about quantum groups may be found on its web page: http://www.math.ku.dk/english/research/conferences/2014/tqft/ They have incidentally a crash course on operator algebras as well, which is part of the theory that allows you to reasonably deal with infinite dimensional representations of the Poincaré group.
How the functor point of view on field theories relate to the "classical" one developed in Faria-Melo a bit fuzzy to me, but I suspect you may find some answers in Segal's article on conformal field theories (http://www.math.upenn.edu/~blockj/scfts/segal.pdf -- a pretty shitty scan but you'll find it in his 60th birth day thing).
Of course this leaves out nitty-gritty computational aspects of the kind a physicists would be able to tell you about, and I have never gotten close enough to what the physicists do to actually wanting to renormalize anything (something you apparently need to do because of self-interacting particles producing diverging integrals). This is definitely a pretty big part of QFT you'll be missing if you don't study the physicists approach as well.
It appears the big unifying idea in any case is that a physical system should be invariant under choice of presentation (gauge) up to a group or automorphisms (gauge transformation) and that this is true for classical systems (Lorentz or Poincaré invariance of space or space-time) as well as quantum systems (other Lie groups acting on a vector bundle of states) and that all of physics are more or less fall out as properties of stuff with the right symmetries. This appears to be what physicists and mathematicians agree on either way, so you can't go wrong studying representations.
Aside from Faria-Melo here are some notes I like to look at:
These notes are pretty explicit about the kind of mathematics they use math.lsa.umich.edu/~idolga/physicsbook.pdf
These notes on Lie groups and representation theory are very good. staff.science.uu.nl/~ban00101/lie2012/lie2010.pdf They come with video lectures. webmovies.science.uu.nl/WISM414
I was donated both volumes of Morse & Feshbach by a retiring professor (who had bought copies forgetting he already owned them!), so let me say a few words about them:
- As noted, the volumes cover a lot of material - inevitably some things are glossed over. Fortunately each chapter ends with an excellent bibliography to chase up those finer details.
- The material is fairly self-contained; it should certainly be accessible to someone with your background in mathematics.
- Problems are also given at the end of each chapter, though I am unaware of any (official) solution book in existence.
- The chapters also end with a summary and/or summarizing table which I have found very useful as a reference. This point (I feel) is quite pertinent: I very much use and feel that M&F is more of a reference book; I have often needed to look elsewhere for the finer mathematical details (which is where the excellent bibliography comes in useful!)
- The books I own are from the 1950s; hence some terminology and the way the material is presented feels a little dated.
I can't comment too much on Courant & Hilbert other than too say that I have found their treatment of some topics to be more rigorous than that of Morse & Feshbach. These books are both more on analytic methods in mathematical physics; you may also be interested to look at applications of group theory and differential geometry to mathematical and theoretical physics, both of which have entire books devoted to them. Let me finish with a couple of alternatives that I have used:
- A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres
- Mathematical Physics: A Modern Introduction to Its Foundations by Sadri Hassani
Best of luck with the studies and hope this helps in some way!
RESPONSE TO COMMENT ON LACK OF PROBLEMS IN H&C:
If you were intent on reading Hilbert & Courant, then the problems in the books already mentioned above can be used to track your progress. There are, however, specific problem books which could be used:
- A Collection of Problems in Mathematical Physics by B. M. Budak et al
- Mathematical Analysis of Physical Problems by P. R. Wallace
- Worked Problems in Applied Mathematics by N. N. Lebedev
- Problems and Solutions in Theoretical and Mathematical Physics by W.-H. Steeb
are ones that spring to mind. The first two are Dover books and are therefore inexpensive; the second one is in two volumes: introductory and advanced. And I am sure there are plenty more!
Best Answer
Honestly, there is no one book which gives you the whole mathematics used in physics and engineering with enough details. If you want to dive deeply into concepts you should read special books for special concepts. However, here are some books similar to Arfken 's book you mentioned where some of them contain more details.
$1$. Mathematical Physics, A Modern Introduction to Its Foundations By Sadri Hassani. This book provides a nice balance between the details and spectrum of the concepts. It is much better than Arfken in my opinion. It tries to convey modern mathematical thinking. The materials are introduced in a smooth integrated way. I think it is a good choice.
$2.$ Mathematical Methods of Physics, Volume 1 and Volume 2 By R. Courant and D. Hilbert. This is a classic which still worth reading as has been written by two well known mathematicians. It provides nice rigor and a vast of concepts.
$3.$ Advanced Calculus for Applications By Francis B. Hildebrand. This is also a classic which is taught in some universities.
$4.$ Mathematical Methods for Engineers and Scientists, Volume 1, Volume 2 and Volume 3 By Kwong Ting Tong. A rather new book which worth looking at.
$5.$ Advanced Engineering Mathematics By Erwin Kreyszig. It is more like an encyclopedia than a book.
$6.$ Fundamentals of Mathematical Physics By Edgar A. Kraut. This one is a popular book in mathematical physics and people are satisfied by reading it. The text is user friendly. You can take a look at reviews in Amazon.
I should emphasize again that none of such books replaces a special book written for a special mathematical subject. They have to sacrifice details for saving brevity. So do not expect these books to provide a systematic step by step rigorous approach.